Tif/.l 




J?iQ_ 2 .tins'* A. S.J'iintiersm 

X. Telescopic -7zew- o£ -fke itijl Mx> on.. • S.TeLescoprc vi.-w of Saturn '-'..is xiug*. 

cLo of a^gmt of the Moon mesa? cp^iaili-atiLL'p '. 4. do of Jiwttpr &■!««. "NTooois. 






AN 



INTRODUCTION 



TO 



ASTRONOMY 



DESIGNED AS A 



TEXT-BOOK 



FOR THE USE OF 



STUDENTS IS COLLEGE. 



BY 







DEOTSON OLMSTED, LL.D., 

PROFESSOR OP ASTRONOMY IN TALE COLLEGE, 
AND 

E. S. SNELL, LL.D., 

PROFESSOR OP MATHEMATICS IN AMHERST COLLEGE. 



THIRD STEREOTYPE EDITION 

Carefully revised, with additions. 









^ 



NEW YORK: 

COLLINS & BROTHER, 

414 BROADWAY. 



V 



Q3* 



3 



JLatared according to Act of Congress, in th^, yet- la-Mt 

By DENISON OLMSTED, 

in tne Clerk's Office of the District Court of Cot lectiofflt 



Revised Edition. 

Entered according to Act of Congress, in the year 18&I, 

By JULIA M. OLMSTED, 

For the Children of Demson Olmsted, deceased, 

Ls 3he Clerk's Office of the District Court of the District of Ckmasettaftg 



Thibd Stereotype Edition. 

Enteied according to Act of Congress, in the year 1866, 

By JULIA M. OLMSTED, 

For the Children of Demson Olmsted, deceased, 

Li iko Clark s Office of the District Court jf the Dfcfcr? * at Coa&eetlMk 



Third Stereotype Edition. 

Carefully revised, with, additions; 

Copyright, 1883, 

By JULIA M. OLMSTED. 



PREFACE TO THE EDITION OF 1883. 



The late discoveries made in Astronomy, principally by 
the aid of the spectroscope, require that something be added to 
the descriptive parte of this work. In the present edition,, 
therefore, information of this nature, accompanied with illus- 
trations, is given in an Appendix, with references to and from 
the corresponding articles in the text. 

The mean equatorial Horizontal Parallax of the Sun, adopted 
from Professor Newcomb's "Investigation of the Distance of 
the Sun and the Elements which depend on it," is 8". 848. This 
number is founded upon a discussion and combination (with 
their relative weights) of the results given by all the different 
methods of obtaining the parallax, and therefore is as near an 
approximation to the truth as can be made at present. The 
distances and magnitudes throughout the work are reduced to 
conform to this value. 

This edition contains the latest emendations of Professor 
Snell ; and also various numerical corrections, in accordance 
with the best authorities, for which the Publishers are indebted 
to Professor Selden J. Coffin, Lafayette College. 

Professor Coffin has also added to Art.. 264, Appendix M, 
and has enlarged and thoroughly revised Tables II, IV, 
and V. 

August, 1883. 



CONTENTS 



CHAPTER I. 

PA«» 

Astronomy.— Its subject. — Globular form of the earth proved.— Modes of 
measuring the earth. — The terrestrial equator. — The horizon and seconda- 
ries. — The celestial equator. — The ecliptic. — The diurnal motion. — Its phe- 
nomena. — Problems on the globes 1-14 

CHAPTER II. 

Parallax. — Diurnal parallax.— Its variation.— To find the parallax of the moon. 
— Atmospheric refraction. — Illumination of the sky. — Twilight 15-28 

CHAPTER III. 

The observatory. — The transit-instrument. — The astronomical clock. — Measur- 
ing right ascension. — The mural circle. — Measuring declination. — Altitude 
and azimuth instruments.— The sextant. — Spherical problems 24-3$ 

CHAPTER IV. 

Observations of the sun's place. — The ecliptic and zodiac. — The annual mo- 
tion. — The change of seasons. — Arrangement of heat and cold. — Form of the 
earth's orbit. — Mode of determining it 88-47 

CHAPTER V. 

The sidereal and solar day. — Mean and apparent solar time. — Reasons why 
solar days are unequal. — The equation of time. — The calendar 47-54 

CHAPTER VI. 

• 

Projectile, centripetal, and centrifugal forces. — Laws of centrifugal force. — Its 
effects on the earth. — Loss of weight. — Spheroidal form. — Proofs of diurnal 
motion 54-63 

CHAPTER VII. 

The sun. — Its form. — Its distance. — Its dimansions. — Its rotation. — Solar spots 
—Theory of spots. — Condition of the sun's surface. — The zodiacal light 62 -68 



CONTENTS. 
CHAPTER VIII. 

PAGB 

Kepler's laws, — Law of areas proved. — Law of gravity proved. — Its prevalence 
throughout the system. — The paths of projectiles. — Effect of an impulse on 
one body of a system 69-8Q 

CHAPTER IX. 

Precession of equinoxes. — Consequent motion of the poles. — Cause. — Compo- 
sition of rotations. — The tropical and sidereal years. — Nutation. — Aber- 
ration of light. — Velocity of light discovered. — Advance of apsides. — Its 
cause. — How to find the sun's true place 80-88 

CHAPTER X. 

The moon. — Its distance and size. — Its motion round the earth. — Its orbit. — 
Librations. — Its path about the sun. — Its phases. — The harvest moon. — 
The moon's surface. — Measurement of its mountains. — Appearance of the 
earth from the moon 88-102 

CHAPTER XL 

The moon's motion disturbed by the sun. — Gravity to the earth diminished. 
— Equations for finding the moon's place. — Equation of the center. — Evec- 
tion. — Variation. — Annual equation. — Advance of apsides. — Eetrogradation 
of nodes. — Periodical and secular equations 102-109 

CHAPTER XII. 

Eclipses. — Their cause. — Eclipse months. — The earth's shadow. — Its dimen- 
sions computed. — To find beginning, middle, and end of a lunar eclipse. — 
Eclipse of the sun. — Dimensions of the moon's shadow. — Its velocity over 
the earth. — The Saros. — Phenomena of a solar eclipse 109-124 



CHAPTER XIII. 

(ftethods of determining longitude. — By the chronometer. — By eclipses. — By 
the lunar method. — By the telegraph. — Change of days in going round the 
earth 125-1 29 

CHAPTER XIV. 

Tides. — Form of equilibrium under the action of the moon. — Joint action of 
sun and moon. — Diurnal inequalities. — Effect of coasts. — Tides in seas and 
lakes 180-185 



CHAPTER XV. 

Planets grouped. — Distances from the sun. — Revolutions. — Dimensions. — 
Masses and densities. — Mercury. — Its motions. — Its phases. — Its transits. — 
Venus. — Its transits. — Parallax of the sun found. — Mars. — Its motions 135-151 



CONTENTS. 
CHAPTER XVI. 

PA6S 

The planetoids. — Jupiter. — Its belts. — Its satellites. — Their eclipses and oo- 
cultations. — The velocity of light found by them. — Saturn. — Its rings. — 
Their disappearances. — The satellites of Saturn. — Uranus. — Its satellites. 
Neptune. — Its discovery 152— Kit 

CHAPTER XVII. 

Moments of a planetary orbit. — Method of finding the first. — The second. — 
The third. — The fourth. — The fifth and sixth. — The masses of the planets 
found. — Perturbations. — In the positions of orbits. — In their forms.— Sta- 
bility of the system. — Relations of the planets 165-180 

CHAPTER XVIII. . 

Comets. — Their number. — Effects of eccentricity of orbit, — Dimensions of 
comets. — Their masses. — How to find their orbits. — Halley's comet. — 
Comets of short period. — A resisting medium. — Remarkable comets. — 
Shooting stars. — Meteoric showers. — Aerolites 180-194 

CHAPTER XIX. 

The stellar universe. — Classifications of stars. — Constellations. — Annual par- 
allax. — Stars whose distance is known. — Nature of fixed stars. — Proper 
motions. — Double stars. — Binary stars. — Their orbits. — Their masses.— 
Periodic stars. — Clusters. — Nebulae. — The galaxy. — The Nebular hypothe 
sis 194-218 

Appendix A to M 214-224 

Tables I.— The Calendar 226-227 

II.— Elements of the Planets 228 

III.— Elements of the Satellites 229 

IV.— Mean Places of Principal Stars 230 

Y.— Planetoids . . . i 231-233 

Plates.— Spectroscope 225 

Chronograph 234 

Comet of 1843. 
Comet of 1858.— Nebulae. 
Part of Galaxy.— Double Stars. 
Clusters.— "Nebulae. 



ASTRONOMY. 



CHAPTEE I. 

GENERAL FORM AND DIMENSIONS OF THE EARTH. — THE 
DIURNAL MOTION. — ARTIFICIAL GLOBES. 

1. General definitions. — Astronomy is the science which 
treats of the heavenly bodies — that is, of the sun, the planets 
and their satellites, the comets, and the fixed stars. 

The sun, planets, satellites, and comets constitute the solar 
system, which is so called because the sun is the principal body 
belonging to it, and controls the movements of all the others. 

The fixed stars are the bodies situated at vast distances out 
side of the solar system, and which, on account of that distance, 
exhibit little or no change of position with respect to each 
other. 

2. The Copernican system. — This name is given, in honor of 
Copernicus, to the science of astronomy as now established by 
demonstration, in distinction from the erroneous systems of the 
ancients. It explains the diurnal and annual motions of the 
heavens, by supposing the earth to rotate each day on its axis, 
and to revolve once a year around the sun. 

3. The globular form of the earth. — That the earth is nearly 
if not exactly a sphere, is indicated in several ways. 

1. It is one of the planets. And, as we see the other planets 
to be nearly spherical, we reason from analogy that the earth is 
6pherical also. 



2 DIP OF THE HORIZON. 

2. Iii a lunar eclipse, whichever side is turned toward the 
moon, the outline of its shadow, projected on that lody, is 
always circular. 

3. Its convexity, by which it wholly or partially conceals 
distant objects, as a lighthouse or a ship at sea, appears to be 
equally great on all parts of the ocean. 

4. An arc of a given number of miles, measured on any part 
of the earth, is found always to subtend an angle of nearly 
equal size at the center ; showing that the curvature is every- 
where nearly the same. 

5. The depression, or dip of the horizon, is equally great at 
every place, and on every side of the observer, provided his 
elevation above the ocean level is the same. This will be un- 
derstood by the next article. 



Fig.l. 



4. Dip of the horizon. — If the eye were at A (Fig. 1) on 
the surface of the earth, the vault of the heavens would be lim- 
ited by a plane touching the earth at 
A, and would therefore be just a hemi- 
sphere. But if the eye is elevated, as 
to O, and tangent lines are drawn from 
that point to the earth on every side, 
then more than a hemisphere of the 
sky is visible. Let ZC be the direc- 
tion of a plumb-line, and let HOR 
represent a plane perpendicular to it ; 
then there would be a celestial hemi- 
sphere in view above this plane, and 
the remotest visible points on the earth 
would be depressed below the plane by 
the angle HOD or ROE. This angle 
is called the dip of the horizon. If AO is a given height, 
it is found that the angle HOD is sensibly equal on whatever 
side of the station, or on whatever part of the earth, the 
measurement is made. It follows from this that the earth is 
very nearly a sphere. 

At the height of 100 feet, the depression is about 10', and 
varies nearly as the square root of the height. 

The word down expresses the direction in which a plumb- 




DIMENSIONS OF THE EARTH. 3 

line hangs, or a body falls — that is, toward the center of the 
earth. Hence, on different parts of the earth, " down" denotes 
all possible directions. So " up," or from the center, is in every 
direction ; and the direction which is down at one place, is up 
at a place on the opposite side of the earth. 



Fig. 2. 



5. Dimensions of the earth. — The semi-diameter of the 
earth may be approximately fonnd by measuring the height oi 
the station AO (Fig. 1), and the 
length of the tangent line OD. If O 
were the summit of a mountain, then 
D would be the most distant point 
from which it could be discerned. In 
Fig. 2, suppose that the height of the 
mountain BD, and the distance to the 
point where it is just seen in the hori- 
zon AD, have been measured. Let 
BD = A, and AD = d, and the radius, 
AC or BO = x. Then a? 2 + d 2 ={x + hy 




■x* + 2 hx + h\ Hence, 2 hx = d 2 — h% and x = 



2A 



Thus, the semi-diameter of the earth is found in terms of k 
and d. 

The magnitude of the earth may be more accurately found, 
by measuring the arc of a meridian. Let a line be carefully 
measured due north on the earth's surface, and the correspond- 
ing difference of latitude be observed, as indicated by the 
change in the elevation of the stars. Then, the surveyed line 
is the same part of the earth's circumference, which the differ- 
ence of latitude is of 360°. Thus, if the arc is 1° 30 r , its length 
is found to be about 103.5 miles. Hence, 

1° 30' : 360° :: 103.5 : 24,840; 
which is nearly the number of miles in the circumference oi 
the earth. By a comparison of the most accurate measure 
ments, it is ascertained that 

The circumference of the earth = 24,857 miles. 

The diameter (24,857 -f- 3.14159+) = 7,912.4 miles. 

One degree of the circumference = 365,000 feet. 

One second = about 100 feet. 



4 SECONDARIES OF THE EQUATOR. 

6. Inequalities of surface. — Although the surface of the 
earth is uneven, and there are high mountains and deep valleys 
in many parts of it, yet these are very minute compared with 
the magnitude of the entire earth ; so that the spherical form 
is not disturbed by their existence. Mountains, four or five 
miles high on the earth, are relatively no more than are the 
particles of dust which adhere to a globe one foot in diameter. 
Thin writing-paper, pasted upon such a globe in the form ot 
the continents, would be sufficiently thick to represent their 
general elevation above the oceans. 

7. The diurnal rotation. — The earth revolves continually 
from west to east, on an imaginary line drawn through its cen- 
ter, called the earth? s axis. The time occupied in completing a 
revolution is called a day, which is divided into twenty-four 
hours. A great circle of the earth, perpendicular to the axis, 
is called the equator. In the diurnal rotation, every particle of 
the earth describes a circle, whose plane is either parallel to 
the equator or coincident with it. The extremities of the axis 
are called respectively the north and south poles. 

8. Secondaries of the equator. — All great circles passing 
through the poles, and therefore perpendicular to the equator, 
are called meridians. Such a circle may be supposed to pass 
through any place whatever on the earth, and is called the me- 
ridian of that place. As all great circles of a sphere which are 
perpendicular to a given great circle, are called its secondaries^ 
the meridians are secondaries of the equator. 

The latitude of a place is its distance north or south from the 
equator, measured on the meridian of that place, in degrees, 
minutes, and seconds. Parallels of latitude are small circles 
of the earth, parallel to the equator. 

The longitude of a place is the distance of its meridian in 
degrees, minutes, and seconds, east or west from ^ome standard 
meridian, as that of the observatory of Greenwich. The people 
of different nations usually reckon longitude from s^Oie import- 
ant observatory of their own country. Thus, the X lench reckon 
from Paris, and the Americans from Washington. Any place 
on the earth is determined by giving its latitude and longitude 



THE HORIZON AND ITS SECONDARIES. 5 

9. The celestial sphere. — The earth is called the terrestrial 
sphere. The celestial sphere is that apparent vault, called the 
sky, which surrounds the earth on every side, and to which all 
the heavenly bodies seem to be attached. The center of the 
earth is regarded as the center of the celestial sphere also. Bat 
the distance of nearly all the heavenly bodies is so immense, 
that it is immaterial from what point of the earth they are 
viewed. Hence, for most purposes of astronomy, the eye of 
the observer may be considered as the center of the celestial 
sphere. 

10. The horizon and its secondaries. — If the plumb-line 
(usually called the vertical), at any place on the earth, is sup- 
posed to be extended till it intersects the celestial sphere, it 
marks the zenith above the place, and the nadir below it. 
And a plane passed through the center of the earth, perpendic- 
ular to the vertical, is called the rational horizon of that place. 
This is a great circle of the celestial sphere, and divides it into 
-upper and lower hemispheres. The sensible horizon is parallel 
to the rational horizon, and passes through the place on the 
earth's surface. The planes of these two horizons are therefore 
near 4,000 miles apart ; but so great is the distance of the 
heavenly bodies, that the two planes seem to unite in the same 
great circle of the heavens. 

If the observer is at all elevated above the earth's surface, 
the boundary line between sky and water is a little lower than 
the horizon, so that somewhat more than half of the celestial 
sphere is in view (Art. 4). The secondaries of the horizon 
intersect each other in the vertical line, and are called vertical 
circles. One of them is the meridian of the place. The inter- 
sections of the meridian and horizon are the north and south 
points of compass. The vertical circle at right angles to the 
meridian is called the prime vertical. This intersects the hori- 
zon in the points called east and west. 

The altitude of a heavenly body is its elevation above the 
horizon, measured on the vertical circle passing through the 
body. The zenith distance of a body is the distance between 
it and the zenith, and is therefore the complement of its 
Altitude. 



6 CELESTIAL EQUATOR. 

The azimuth of a heavenly body is an arc of the horizon > 
measured from the meridian to the vertical circle, which passes 
through the body. The amplitude is measured from the verti 
cal circle passing through the body to the prime vertical, and 
is therefore the complement of the azimuth. The altitude, or 
zenith distance of a heavenly body, along with its azimuth or 
amplitude, determines its place in the visible heavens. 

1 1 . The celestial equator and its secondaries. — If the axis on 
which the earth revolves is produced to the heavens, it becomes 
the axis of the celestial sphere, and marks the north and south 
poles of that sphere. The north pole is at present in the con- 
stellation of Ursa Minor. If the plane of the equator be ex- 
tended in like manner, it becomes the celestial equator. The 
secondaries to this circle are called meridians, as on the earth* 
They are also called hour-circles, because the arcs of the 
equator intercepted between them are used as measures of 
time. 

Fig. 3. 



ZD 




Let n (Fig. 3) represent the north pole of the earth, s its 
bo nth pole, eqthe equator (projected in a straight line), o a given 



THE ECLIPTIC. 7 

place whose north latitude is eo. Then N, S, are the poles of 
the celestial sphere, EQ is the celestial equator, Z is the zenith 
of the place o, R is its nadir, and HO its rational horizon. 
oesqn is the terrestrial meridian of the same place, and 
ZESQK is its celestial meridian, or hour-circle. 

1 2. The ecliptic. — Besides the equator, there is an import- 
ant circle of the celestial sphere, called the ecliptic. It is that 
in which the sun appears to make its annual circuit around the 
heavens. It is inclined to the equator at an angle of nearly 
23J°, crossing it in two opposite points, called the equinoctial 
points, or equinoxes. The word " equinoxes" is used also to 
express the times at which the sun crosses the equator, because 
at those times the nights are equal to the days. The vernal 
equinox is the time when the sun passes the equator from south 
to north, as it occurs in the spring, about March 20th. The 
autumnal equinox occurs on or near September 22d, when the 
sun returns to the south of the equator. 

The solstitial points, or solstices, are those points of the 
ecliptic, which are furthest north or south from the equator, 
situated therefore midway between the equinoxes. They are 
so named, because there the sun stops in his advance north- 
ward or southward, and begins to return. The summer solstice 
is the point where, and also the time when the sun is furthest 
north, about the 21st of June. He passes the winter solstice on 
or near the 21st of December. 

The equinoctial colure is that secondary to the equator 
which passes through the equinoxes. The solstitial colure is 
that which passes through the solstices. They are therefore at 
right angles to each other, and the latter is a secondary to the 
ecliptic, as well as to the equator. 

13. Signs of the ecliptic— The ecliptic is divided into 
12 equal parts of 30° each, called signs, which, beginning at 
the vernal equinox, succeed each other eastward, in the follow 
ing order : 



DIUKNAL MOTION OF THE HEAVENS. 



Northern. 




Southern. 




1. Aries . . 


. f 


7. Libra . . , 


=Cb 


2. Taurus . . 


. 8 


8. Scorpio 


m 


3. Gemini . . 


n 


9. Sagittarius 


* 


4. Cancer . . 


. © 


10. Capricornus 


V3 


5. Leo . . . 


. $1 


11. Aquarius . 


AW 


6. Yirgo . . 


. T02. 


12. Pisces . . 


X 



The vernal equinox being at the first point of Aries, the sum 
mer solstice is at the first of Cancer, the autumnal equinox at 
the first of Libra, and the winter solstice at the first of Capricorn. 

14. R'ght ascension and declination. — The right ascen- 
sion of a heavenly body is the angular distance of its meridian 
from the vernal equinox, measured eastward on the equator. 
The declination of a body is its angular distance north or south 
from the equator, measured on the meridian of the body. 

The equator is the plane of reference for right ascension and 
declination on the celestial sphere, as it is for latitude and 
longitude on the terrestrial. But terrestrial longitude is reck- 
oned both east and west, while right ascension is reckoned only 
to the east. 



15. Celestial longitude and latitude. — On the celestial 
sphere, longitude and latitude are referred to the ecliptic, not 
to the equator. Suppose a secondary to the ecliptic to pass 
through a heavenly body ; the distance of the body from the 
ecliptic, measured on the secondary, is its latitude ; and the dis- 
tance of this secondary from the vernal equinox, measured 
eastward on the ecliptic, is its longitude. 

Eight ascension and longitude are reckoned only eastward, 
from 0° to 360°, the first on the equator, the other on the 
ecliptic. 

- 1 6. Apparent diurnal motion of the heavens. — As the earth 
revolves from west to east on the axis ns, an observer, not 
being conscious of this motion, sees the heavenly bodies appa- 
rently revolving in the opposite direction — that is, from east to 
west, about the axis NS. The sun, moon, and every planet, 



comet, and star, is observed to pass over from the eastern part 
of the sky toward the western, with a regular motion, reap- 
pearing again in the east, after the lapse of about one day, in 
the same, or nearly the same place. The fixed stars describe 
circles, which are exactly parallel to the equator, and in pre- 
cisely the same length of time. But the other bodies vary 
somewhat in their paths, and the periods of describing them, 
thus indicating that they are affected by other motions besides 
the diurnal rotation. 

17. Rising, setting, and culmination. — In Fig. 3, AB, 
DO, FG, etc., drawn parallel to EQ, represent the diurnal 
circles of stars, projected in straight lines. Some of these 
circles intersect the horizon HO. These intersections are the 
points of rising or setting. Thus, a star describing the circle 
GF, rises in the northeast quartei, and sets in the northwest, 
at points which are both represented by r. The star, whose 
diurnal circle is IK, rises in the southeast, and sets in the south- 
west, at t. A star on the equator rises exactly in the east, and 
sets in the west, at the point G. 

The points, in which these circles cut the meridian, are 
called the points of culmination. Thus, the star on FG makes 
its upper culmination at F, arid its lower one at G. On AB, 
both the upper and lower culminations are above the horizon ; 
on MP, they are both below. If both culminations of a star are 
above the horizon, it is always in view; if both below, it never 
comes in sight. The number of stars which do not rise and set, 
depends on the position of the celestial poles in relation to the 
horizon — that is, on the latitude of the place. 

By the culmination of a body, in the ordinary use of the 
word, is meant its upper culmination. 

18. Relations of the horizon to the diurnal circles. — 
Every change of position on the earth changes the horizon. If 
an observer moves eastward, all the heavenly bodies which rise 
and set, rise earlier, and also culminate and set earlier. If he 
moves westward, they rise, culminate, and set later. If he 
moves toward the nearer pole of the earth, the corresponding 
pole of the celestial sphere becomes more elevated, and the 



10 THE PARALLEL SPHERE. 

other more depressed ; and the contrary, if he moves from the 
nearer pole — that is, toward the equator. In all north latitudes, 
the north pole is elevated, and the south pole depressed ; and 
the reverse in south latitudes. And the elevation of one pole, 
and the depression of the other, equals the latitude. For 
(Fig. 3) NO, the elevation of one pole (=HS, the depression 01 
the other), equals EZ, since each is the complement of ZN 
But EZ=&9, the latitude, because they subtend the same angl 
atC. 

The elevation of the celestial equator equals the complement 
of latitude. For EH is the complement of EZ, which equals 
eo, the latitude. Hence, the angle by which all the circles of 
diurnal motion are inclined to the plane of the horizon, equals 
the complement of latitude, since they are parallel to the 
equator. 

On account of this change of inclination between the horizon 
and the diurnal circles, the aspect of the diurnal rotation is 
very different in different parts of the earth. 

19. Tht right sphere. — This name is given to those posi- 
tions, in which the diurnal circles cut the horizon at right 
angles. All points of the equator are so situated. As the 
latitude is zero, the poles, having no elevation or depression 
(Art. 18), are both in the horizon ; the celestial equator passes 
through the zenith, thus coinciding with the prime vertical; 
and all the paths of daily motion, being parallel to the equator, 
are perpendicular to the horizon. Every heavenly body, unless 
situated exactly at one of the poles, rises and sets during each 
revolution, and continues above the horizon just as long as it 
remains below it. If a star rises in the east, it sets in the west, 
and culminates in the zenith and nadir. 

20. The parallel sphere. — This term expresses the appear- 
ance of the heavens at those points of the earth where the 
circles of daily rotation are parallel to the horizon. This aspect 
can be presented only at the poles. For, at those points, the 
latitude being 90°. one pole must be elevated 90° — that is, to the 
zenith — and the other depressed 90°, or to the nadir. Hence. 
U?e diurnal circles, being perpendicular to the axis, must Iks 



ARTIFICIAL GLOBES. 11 

horizontal, and the equator must coincide with the horizon. 
Every star in view passes around the sky, maintaining the 
same elevation at every point of its path. JSTo one of the iixed 
stars ever rises or sets, and every point of a diurnal circle may 
be regarded as a point of culmination, since it is on a meridian 
passing through the observer's place. 

At the north pole, that half the year in which the sun is 
north of the equator, is uninterrupted day ; during the other 
half, the sun being south of the equator, it is constant night. 

In the right sphere, the whole sky is seen, and every part of 
it just half the time ; in the parallel sphere, only one-half the 
sky is ever seen, but it is seen the whole time. 

2 I . The oblique sphere. — At all latitudes, except 0° and 
90°, the circles of daily motion are oblique to the horizon, since 
they incline at an angle equal to the complement of the lati- 
tude. Thus, at latitude 42° N., the celestial equator is elevated 
48° above the southern horizon, and all the diurnal circles have 
the same inclination, as shown in Fig. 3. The circle OD, 
whose distance from the elevated pole equals its elevation, just 
touches the horizon at the lower culmination, and is the limit 
of that part of the sky which is always in view. This is called 
the circle of perpetual apparition. The circle HL, at the same 
distance from the depressed pole, also touches the horizon, and 
is called the circle oft. perpetual occultatio?i, since it limits that 
part of the sky which is always concealed. 

The horizon HO, bisects the equator EQ. Hence, a body 
on the equator is as long above the horizon as below it, in every 
part of the earth. But all bodies between the equator and the 
elevated pole are longer above the horizon than below, while 
on the opposite side they are longer below than above. 

22. Artificial globes. — They are of two kinds, terrestrial 
and celestial. The terrestrial globe is a miniature representa- 
tion of the earth, having also the equator and several meridians 
and parallels of latitude traced upon it. The celestial globe 
exhibits the principal fixed stars in their relations to each 
other, and to the equator and ecliptic. 

The artificial globe is suspended in a strong brass ring by an 



Cl PROBLEMS ON THE GLOBES. 

axis passing through the north and south poles, Dn which it is 
free to revolve. This ring represents the meridian of any place, 
and is supported vertically within a horizontal wooden ring 
which stands upon a tripod. The wooden ring represents the 
horizon. The brass ring may be slid around in its own plane, 
so as to elevate or depress either pole to any angle with the 
horizon. It is graduated from the equator each way to the 
poles, for measuring latitude and declination ; while the horizon 
ring has near its inner edge two graduated circles, one for 
azimuth, and the other for amplitude. On this ring also, for 
convenient reference, are delineated the signs of the ecliptic, 
and the sun's place in it for every day of the year. 

Around the north pole is a small circle, marked with the 
hours of the day ; and at the same pole, a brass index is attached 
to the meridian, which can be set at any hour of the circle. 

The quadrant of altitude is a flexible strip of brass, graduated 
into 90 parts, each equal to a degree of the globe. This can 
be used for measuring angular distances in any direction on the 
sphere ; and when applied to a vertical circle of the celestial 
globe, it determines the altitude, or zenith distance of a heav- 
enly body. 

To adjust either globe for any place on the earth, elevate the 
corresponding pole to a height equal to the latitude. The axis 
will then form the proper angle with the horizon. And if the 
globe is turned (the celestial westward, or the terrestrial east- 
ward), the diurnal motion will be truly represented. 



23. Problems on the terrestrial v 

1. To And the latitude and longitude of a place. 

Turn the globe so as to bring the place to the brass 
meridian ; then the degree and minute on the meridian 
over the place shows its latitude, and the point of the 
equator, under the meridian, shows its longitude. 

Example. "What are the latitude and longitude of 
New York? 

2. To find a place by its given latitude and longitude. 

Find the given longitude on the equator, and bring 
it to the meridian; then under the meridian, at the 
given latitude, will be found the required place. 



PROBLEMS ON THE GLOBES. 13 

Ex. What place is in latitude 39° 2SL, and longitude 
77° W. ? 

3. To find the beaiing and distance of one place from 
another. 

Adjust the globe for one of the places, and bring it 
to the meridian ; screw the quadrant of altitude directly 
over the place, aud bring its edge to the other place. 
Then the azimuth will be the bearing of the second 
place from the first, and the number of degrees between 
them, multiplied by 69J, will give their distance apart 
in miles. 

Ex. Find the bearing of New Orleans from New 
York, and the distance between them. 

4. To find the difference of time at different places. 

Bring to the meridian the place which lies west of the 
other, and set the hour-index at XII. Turn the globe 
westward, until the other place comes to the meridian, 
and the index will show the hour at the second place 
when it is noon at the first. The hour thus found ie 
the difference required. 

Ex. When it is noon at New York, what time is it 
at London ? 

5. The hour being given at any place, to find what hour 
it is at any other place. 

Find the difference of time between the two places, 
as in (4) ; then, if the place, whose time is required, is 
east of the other, add this difference to the given time ; 
but if west, subtract it. 

Ex. What time is it in Boston, when it is 2 p. m. in 
Paris ? 

6. To find the antiscii, the perioeci, and the antipodes of a 
given place. 

Bring the given place to the meridian ; then, under 
the meridian, in the opposite hemisphere, in the same 
degree of latitude, are found the antiscii. Set the 
index to XII., and turn the globe until the other XII. 
is under the index ; then, the perioeci will be at the 
same point of the meridian as the given place was, and 
the antipodes will bo where the antiscr. wpra 



14 PROBLEMS ON THE GLOBES. 

Ex. Find the antiscii, the perioeci, and the antipodes 
of Lake Superior. 

To the antiscii, the hour of the day is the same as at 
the given place, but the season is reversed. To the 
perioeci, the season is the same, but the hour opposite. 
To the antipodes, both hour and season are opposite 

24. Problems on the celestial globe. 

1. To find the right ascension and declination of a heav 
enly body. 

Bring the place of the body to the meridian ; then 
the point directly over it shows its declination ; and the 
point of the equator under the meridian, its right 
ascension. 

Ex. Find the right ascension and declination of a 
Lyras. Also, of the sun on the 3d of May. 

2. To represent the appearance of the heavens at any time. 

Adjust the globe for the place. (Art. 22.) On the 
wooden horizon find the day of the month, and against 
it is given the sun's place in the ecliptic. On the 
ecliptic find the same sign and degree, and bring the 
point to the meridian. The globe then presents the 
positions of the stars at noon. Set the hour-index at 
XII., and turn the globe till the index points to the 
required hour. The aspect of the heavens at that hour 
is then represented. 

Ex. Required the aspect of the stars at Lat. 51°, Dec. 
5th, at 10 p. m. 

3. To find the time of the rising and setting of any heav- 
enly body, at a given place. 

Having adjusted for the latitude, bring the sun's 
place in the ecliptic to the meridian, and set the index 
at XII. Turn the globe eastward, and then westward, 
till the given body meets the horizon, and the index 
will show the times of rising and setting. 

The times of the surfs rising and setting may be 
found in the same manner, on the terrestrial globe, 
since the ecliptic is usually represented on it. 



PARALLAX DEFINED. 15 

Mb. At what time does the sun rise and set on the 
4th of July? 

Find the time of the rising and setting of Arcturua 
on the 10th of November. 
i. To find the altitude and azimuth of a star for a given 
latitude and time. 

Adjust the globe for the aspect of the heavens (2) 
screw the quadrant of altitude to the zenith, and direct 
it through the place of the star. Then, the arc between 
the star and the horizon is the altitude ; and the arc of 
the horizon between the quadrant of altitude and the 
meridian, is the azimuth. 

Ex. Find the altitude and azimuth of Sirius, Dec. 
25th, at 9 p. m. Lat. 43°. 
ft To find the angular distance between two stars. 

Lay the quadrant of altitude across the two stars, so 
that the zero shall fall on one of them ; then, the degree 
at the other will show their distance from each other. 

Ex. Find the distance between Arcturus and a Lyrse. 
6. To find the sun's meridian altitude for a given latitude 
and day. 

Find the sun's place, and bring it to the meridian. 
The degree over it will show its declination. If the 
declination and latitude are both north or south, add 
the declination to the co-latitude ; if not, subtract it. 

Ex. Find the sun's meridian altitude at noon, Aug. 
1st. Lat. 38° 30' K 



CHAPTER II. 

PARALLAX. — ATMOSPHERIC REFRACTION. — TWILIGHT. 

25. Parallax defined. — When a person changes his place, 
objects about him in general appear in different directions from 
him. This change of direction is called parallax. If, for ex- 
ample, he moves north, an object, which was directly west or 



16 



DIURNAL PARALLAX. 



him, is moved by parallax towards the southwest • and an 
object which was east, now appears in the southeast quarter. 
The direction of every thing is more or less altered, except 
those objects which are in the line of his motion. 

26. Diurnal parallax. — While a person therefore travels 
over the earth, or is carried about it by the diurnal rotation, 
the heavenly bodies mnst in the same way suffer some paral- 
lactic change. 

By the true place of a heavenly body, is meant that which it 
would seem to occupy if viewed from the center of the earth. 
At the surface, therefore, it appears generally displaced from 
its true position ; and this displacement is called the diurnal 
parallax. Thus, the true place of the body M (Fig. 4.), is in 
the direction CK ; but at A it appears in the line AH ; and the 
parallax is the angle AMC. 



So, the true place of M' is Q, 
its apparent place is P, and 
the parallax is AM'C. But 
the body W" appears at Z, 
whether viewed from A or C, 
and the parallax in this case is 
zero. Since the earth's radius, 
in each instance, subtends the 
angle of parallax, we have the 
following definition : 

The diurnal parallax of a 
body is the angle at that body 
subtended by the semi-diameter of the earth. 



Fig. 4. 




27. On what diurnal parallax depends. — In the triangle 
ACM', let AC=r, CM'=^, and the parallax, AM'C=p. Let 
the zenith distance of the body, ZAM' = z ; then, the angle 
CAM' is the supplement of z. Hence, 

sin^> : sins :: r : d: 
r sin z 

... smi , = ___ 

Since p is always very small, sinj? varies nearly as^? itsell 



PARALLAX OF THE MOON. 17 

Therefore, regarding r as constant, p <x —-7—. That is, The 

parallax of a tody varies directly as the sine of its zenith 
distance, and inversely as its distance from the earth? s center, 

28. Horizontal parallax. — The largest diurnal parallax, 
which a body can have, occurs when the body is seen in the 
horizon, as at M. It is then called horizontal parallax. From 
the horizon to the zenith, the parallax diminishes through all 
values to zero. 

In the case of a given body, d> is usually constant ; and if its 
parallax, at a certain elevation, has been obtained, its horizontal 
parallax is found by the variation, p oo sin z. At the horizon, 
s = 90°, and sin z = rad. If, when the zenith distance is 53°, 
the moon's parallax is found by observation to be 45', then 
sin 53° : rad : : 45' : 56' 21", which is its horizontal parallax. 

29. To correct for parallax. — The effect of parallax is to 
cause a body to appear lower than its true place. Hence, the 
true altitude of a body is obtained by adding the parallax to 
its apparent altitude. 

As parallax is a depression on a vertical circle, then, if a 
body is on the meridian, the parallax affects its declination just 
as much as its altitude, since the meridian is also a vertical ; bu 
in other cases, the vertical circle being oblique to the equator 
the parallax can be resolved into two components, one of which, 
parallel to the equator, is parallax in right ascension ; the other 
perpendicular to the equator, is parallax in declination. 

30. To find the parallax of the moon. — Let A and B 
(Fig. 5) be two stations on the same meridian, taken as far 
apart as possible. The latitude of each place being known, the 
arc AB — that is, the angle ACB — is known. When the moon 
crosses the meridian, let its zenith distance be observed at each 
station. The observer A sees the moon projected in the sky at 
Y, and the zenith distance is the angle ZAY, while that at B 
is Z BY'. The supplements of these angles, MAC, MBC, are 
therefore known. In the isosceles triangle ABC, obtain the 
angles A and B, and the side AB ; subtract the angles from 



18 



ATMOSPHEEIC REFRACTION. 



Fig. 5. 



MAC and MBC respectively, then MBA, MAB are known, 
which, with the side AB, will give AM and BM. Finally, in 
the triangle AMC, the angle A and sides including it will fur- 
nish the angle AMC, which is the parallax sought for the 
station A, at the zenith 
distance ZAY. From 
this the horizontal paral- 
lax can be obtained, as in 
Art. 28. 

The horizontal paral- 
lax of the moon is much 
greater than that of any 
other heavenly body. Its 
mean value is about 57', 
and is correctly repre- 
sented by the angle 
EMC, in Fig. 6. 

The above method has 
also been employed for 
two or three of the 
planets, when they come near to the earth. But, with these 
exceptions, all the heavenly bodies are so far from us, that their 
horizontal parallax is too small to be obtained in this way with 
sufficient accuracy. The parallax of the sun is less than 9" ; 
that of nearly all the planets is much smaller than this ; and as 
to bodies outside of the solar system, they afford not the 
slightest indication of any diurnal parallax. 




E 



Fig. 6. 



M 



31. Atmospheric refraction. — Before the true place of a 
body can be found by observation, a correction must also be 
applied for the refraction of its light by the atmosphere. While 
parallax depresses bodies below their true places, more or less 
according to their distance, refraction elevates them, the near 
and the distant alike. 

The earth's atmosphere may be conceived to consist of an 



ATMOSPHERIC REFRACTION. 



19 



indefinite n amber of strata, bounded by spherical surfaces, as 
AA, BB, etc. (Fig. 7), these strata being more dense according 
as the j are nearer the earth. Light from a star S, entering the 
air at a, is bent toward the perpendicular to its surface (which 

Fig. 7. 




is the earth's radius produced to that point), and describes ab f 
instead of ax. For the same reason, it is again bent into ho, 
and then into cO ; and therefore the star appears in the direc- 
tion of cO produced, at S', higher than its true place. The 
path of the ray from a to O is in reality not a broken line, as 
in the figure, but a curve, because the changes of density occur 
at every point. A body at the zenith is not moved out of 
place, because its light strikes the surfaces perpendicularly. 
The refraction at the horizon is about 35'. This is the greatest 
of all, since the angle of incidence there is the greatest possible. 
From the zenith to the horizon the refraction constantly in 
creases, — slowly at great elevations, but very rapidly near the 
horizon, as shown in the following table. 



Elevation. 


Eefraction. 


Elevation. 


Eefraction. 


90° 


0' 0" 


20° 


2' 37" 


80 


10 


10 


5 16 


60 


33 


5 


9 47 


45 


58 


2 


18 09 


40 


1 09 


1 


24 25 


30 


1 40 





U 54 



The true size of the largest angle of refraction is seen in 



20 METHODS OF MEASURING REFRACTION. 

Fig. 8. AB is a portion of the surface of the earth, ah the 
surface of the atmosphere, AC, BC portions of the radii of the 
earth ; S is the true place of a star, S' the place as elevated by 
horizontal refraction. 

Fig. 8. 




32. Measurement of refraction, — At latitudes greater 
than 45°, stars which culminate in the zenith make their 
lower culminations above the horizon. Such a star is observed 
at both culminations, and its distance from the pole is measured 
at each. These polar distances are really equal, but appa- 
rently unequal, because below the pole the star is elevated by 
refraction, while at the zenith it is not displaced. The differ- 
ence of the apparent polar distances, therefore, gives the 
amount of refraction at the place of lower culmination. 

The refraction within several degrees of the zenith is so 
slight, and its change so uniform, that observations may be 
made in the same way on stars which culminate several degrees 
north or south of the zenith ; and thus, by applying a small 
correction, the refraction may be measured at many different 
altitudes. 

33. General method of measuring refraction. — A star, 
vrhose declination is known, may be used for determining re- 
fraction at any altitude, in the following manner. 

Let m n (Fig. 9) be the path of diurnal rotation of a star, 
whose declination xr is known. When the star is at x, let its 
apparent altitude be measured, and let the exact time also be 
observed. When it culminates at m, observe the time again. 
The difference of these times, allowing 15° for an hour, will 
give the angle at the pole ZPx. The co-latitude of the place, 
ZP, and the co-declination of the star, P#, being known in the 



TABLES OF REFRACTION. 



21 



spherical triangle ZP#, the side Za? can be computed. Its 
complement xy is the true altitude. This, subtracted from the 
apparent altitude before observed, gives the refraction at that 
elevation. 

Fig. 9. 




34. Tables of refraction. — It is demonstrated, that except 
near the horizon, the mean refraction varies as the tangent ol 
the zenith distance. Tables of atmospheric refraction are cal- 
culated in accordance with this law, for all zenith distances 
less than 80°. They are, however, extended beyond that limit 
down to the horizon, being calculated for the last 10° by a 
different and more complex law, and the results of calculation 
being more uncertain. On this account, all astronomical 
measurements are made, so far as is possible, within 75° of the 
zenith. In order to obtain the place of a body with the utmost 
accuracy, tables of refraction are accompanied with means of 
correcting for the state of the barometer and the thermometer 
at the time of observation. 



35. Time of rising and setting affected by refraction. — 
Since any heavenly body at the horizon is considerably elevated 
by refraction, it therefore appears to rise earlier and set later 



22 TWILIGHT. 

than it would do if there were no atmosphere. The angular 
breadth of the sun is about 32 ', while horizontal refraction is a 
little more than this— 35'. Therefore, the sun appears just 
above the horizon, when, in truth, it is wholly below. This 
adds at least four minutes to the day, two in the morning and 
two at evening. 

36. Distortion of the sun's and moon's disk by refrac- 
tion. — The change in the amount of refraction is so rapid near 
the horizon, that when the sun has just risen, or is just about to 
set, the lower limb is elevated more than the upper, by a very 
perceptible quantity. Its form, therefore, does not appear cir- 
cular, but nearly elliptical, the vertical diameter being shortened 
about 5 7 or 6'. The lower half, however, appears more flat- 
tened than the upper half, because the difference of refraction 
between the lower limb and the center is greater than that, 
between the center and the upper limb. 

37. Illumination of the shy. — During the day, the atmos- 
phere is illuminated by the light of the sun, which penetrates 
every part of it, and is reflected in all directions. If there were 
no air, the sky, instead of appearing luminous by day, would 
exhibit the same blackness as by night, and the stars would be 
visible alike at all times. We should, in that case, lose a great 
part of that generally diffused light which illuminates the 
interior of buildings, and other places screened from the direct 
rays of the sun. The earth's surface, and all terrestrial objects, 
on which the sunlight falls directly, would indeed, by radiant 
reflection, cause a degree of illumination, but it would be far 
less than we now enjoy. It has been observed, in ascending to 
great heights, either on mountains or in balloons, where, of 
course, the air which is most dense and reflects most abun- 
dantly is left below, that the sky assumes a very dark hue, and 
the general illumination is greatly diminished. 

38. Twilight. — The illumination of the sky begins before 
the sun rises, and continues after it sets : it is then called twi- 
light. More or less of it is visible, as long as the sun is not 
more than 18° vertically below the horizon. Those parts of the 



DtrBATION OF TWILIGHT. 



23 



atmosphere are most luminous, which lie nearest to the direc« 
tion of the sun. Thus, in Fig. 10, let A be a place on the 
earth, where the sun is just setting. The whole sky, IEFH, is 
illuminated. But, to a place further east, as B, the twilight 
extends from E to H, — the part of the sky, IIK, remote from 
the sun, being in the shadow of the earth. At C, only FH is 
illuminated, and HL is dark. At D, the twilight is entirely 
gone. 

Fig. 10. 




Though the twilight terminates at H, there is no abrupt 
transition from light to shade at that point, since the reflection 
from those high and rare parts of the air is exceedingly feeble ; 
and also, because the thickness of the illuminated segment, 
through which we look, diminishes gradually to that limit, as 
is obvious from an inspection of the figure. 



39. Duration of twilight. — To an observer at the equator, 
at those times of the year when the sun is on the celestial 
equator, the twilight continues lh. 12m. For, in the diurnal 
motion, 15° are described in an hour, and therefore 18° in 
l T 3 jh. = lh. 12m. This is the shortest duration possible. For, 
if the sun were on a parallel of declination, the degrees of diurnal 
motion would be shorter than those on a great circle. And, 
if the observer were on some parallel of latitude, the circles of 
daily motion would be oblique to his horizon, and the sun must 
therefore pass over more than 18°, in order to move 18° verti 
eally. An extreme case occurs at the poles, where twilight 
lasts several months. 



24 



THE TRANSIT INSTRUMENT. 



CHAPTEE III. 

THE OBSERVATORY AND ITS INSTRUMENTS.- 
PROBLEMS. 



-SPHERICAL 



40. The observatory. — Accurate knowledge of the motions 
of the heavenly bodies is mostly obtained by observing their 
relations to the diurnal rotation. The observatory is furnished 
with several instruments by which such observations are made. 

41. The transit instrument. — This is a telescope so mount- 
ed as to observe a heavenly body, at the instant when it cul- 
minates^ — that is, makes a transit of the meridian. AD 




(Fig. 11) represents the telescope supported by a horizontal 
axis, which consists of two hollow cones, placed base to base, 
so as to combine lightness and strength. The ends of the axis 
rest in sockets, Attached to two stone piers, E and W. That 



ADJUSTMENT OF TRANSIT INSTRUMENT. 



25 



the instrument may receive no tremors from the building, the 
piers stand on a firm foundation in the ground, passing through 
the floor without contact. The axis being placed east and 
west horizontally, the telescope, which is perpendicular to it, 
will, when turned, revolve in the plane of the meridian. A 
graduated circle, n, is attached to one end of the axis, for 
marking altitudes or zenith distances. The whole instrument 
can be raised from the sockets, and the axis inverted, so that 
the east end shall rest on the pier W, and the west end on the 
pier E. 




42. Adjustments of the transit instrument. — The visual 
axis of the telescope, AD, is called the line of collimation^ and 
is marked by the intersection of two exceedingly fine wires in 
the focus of the eye-glass. One of these wires is horizontal, fh 
(Fig. 12), the other vertical, d e ; the latter visibly marks the 
direction of the meridian, when the instrument has been prop- 
erly adjusted. The sockets, in which the ends of the axis 
rest, are so connected with the stone piers, that one of them can 
be raised or lowered by a 
screw, and the other can, in 
a similar manner, be moved 
north or south. By the 
spirit-level, L, which hangs 
on the axis, it can be seen 
whether the axis is horizon- 
tal. If not, raise or lower 
the end which admits of 
vertical motion. To find 
whether the line of collima- 
tion is perpendicular to the 
axis of revolution, observe 
whether a distant terrestrial 
object, which is on the vertical wire, remains on it after the 
ends of the axis have been inverted in their sockets. If not, 
move the plate which carries the wires laterally, till the vertical 
wire bisects the distance between the two positions of the 
object. And finally, to determine whether the axis is east and 
west, observe if a circumpolar star occupies the same length oi 





26 TO OBSERVE RIGHT ASCENSION. 

time in passing from the upper to the lower culmination, as 
from the lower to the upper ; and if not, move the end of the 
axis horizontally, till the intervals are equal. 

For fuller instructions on adjustment, see Loomis's Practical 
Astronomy. 

43. The astronomical clock. — The transit instrument marks 
the event of crossing the meridian ; the clock must be used in 
connection with it, to fix the time of the transit. The clock of 
the observatory is made to keep sidereal time, — that is, it marks 
off 24 hours in the interval between two successive transits of a 
star, instead of the sun. This interval is called a sidereal day, 
and is about 4 minutes less than a solar day. The sidereal day 
begins when the vernal equinox transits the meridian. At that 
instant, the clock is at Oh. Om. Os. ; and any hour of the clock 
shows how long a time has elapsed since the equinox culmi- 
nated. 

44. Error and rate of clock, — The uniform movement of 
the clock is its most important excellence. This may be tested 
by the transit instrument, and a list of right ascensions of stars. 
If it does not indicate Oh. Om. Os. when the vernal equinox cul- 
minates, the difference is called its error. If it marks any more 
or less than 24 hours between two successive transits of a star, 
this gain or loss is called its rate. If both error and rate are 
known, then the true time is known ; and generally it is not 
best to alter the clock, but only to keep a record of error and 
rate. 

45. To observe the right ascension of a heavenly body. — 
Having elevated the telescope to the altitude of the body at the 
time of culmination, notice the exact instant when it appears 
on the vertical wire de (Fig. 12). This is its right ascension, 
which may be given either in time or in arc. Thus, if the 
clock is at 13h. 46m. 32s. when a star passes the wire, its right 
ascension is 13h. 46m. 32s. ; or, at the rate of 15° for each 
hour, 206° 38' 0". 

To secure greater accuracy, several equidistant wires are 
placed parallel to de, an equal number on each side, as in Fig 



THE CHKO.N0GKAPH. 27 

12. The time of passing each wire is noted, and the average of 
all obtained for the time of crossing the central one. 

To observe the right ascension of the snn or a planet, the 
transit of each limb must be noticed, and the mean of all the 
times will be the right ascension of the center of the disk. 

In order to render the wires visible by night, the field of 
view is faintly illuminated by a lamp, placed at one end of the 
hollow axis, the light of which, after entering the telescope, is 
reflected toward the eye-piece. 

46. Transits recorded by the chro?iograph. — To observe the 
time of a star-transit, the eye must discern the instant of its 
bisection by the wire, and the ear must hear the beat of the 
clock, — the seconds being counted from the last completed 
minute before the observation began. If the bisection occurs 
between two beats, as it commonly does, the observer needs 
much practice to be able to divide the second accurately into 
tenths, and decide at which of them the transit takes place. 
Transits are now generally observed and recorded with much 
greater ease and accuracy by the use of the galvanic circuit. 

Fig. 13. 



The pendulum of the observatory clock is arranged to close 
the circuit of a battery and break it again, at the beginning of 
every beat. The closing of the circuit gives a small lateral 
motion to the registering pen, under which the paper is ad- 
vancing on a revolving cylinder, about an inch per second. 
Thus the seconds are all permanently recorded by notches one 
inch asunder in a straight line, as a, b, c, d (Fig. 13). The 
mark at the beginning of each minute has some peculiarity by 
which it may be distinguished from the rest. The observer has 
under his hand a key, which, by a quick touch, will also close 
and break the circuit. Whenever a star is on one of the wires 
of the transit instrument, he touches the key, the pen is moved 
aside, and indents the line as at A, and the observation is thus 
recorded ; and the place where this motion commenced between 
the second-marks can afterward be carefully examined. Thus, 



28 



THE MUEAL CIECLE. 



without the distraction of attending to the clock, he can record 
the transits of all the wires ; and if he only notices within what 
minute the work begins, he can read the entire record with 
accuracy to the T ^ or even the too of a second. Since the 
general adoption of this method, the number of wires has been 
increased, sometimes to 30 or 40, so as to obtain the mean of 
more numerous observations on the same star. The instru 
ment, as above described, is known as the chronograph. 

47. The mural circle. — The circle of the transit instrument 
is used principally for finding a body whose altitude is known, 
and is too small for accurate measurement of arcs on the meri* 



Fig. 14. 




uian. For measuring meridian arcs, the mural circle is em- 
ployed ; so called, because it revolves by the side of a vertical 
wall. It consists of a circle usually six or eight feet in diame- 
ter, and a telescope attached to its face. It is made so large, in 



THE VERNIER. 



29 



order that very small angles may be measured by the divisions 
on its limb. Fig. 14 represents the instrument attached to the 
meridian wall. Its radii are hollow and of conical form. The 
axis, which is on one side only, is firmly set in the wall ; and 
the circle and telescope revolve upon it. The graduations are 
made on the rim, and not on the face of the circle, and are read 
by means of microscopes attached to the wall. 

48. Subdivisions of the graduated limb. — The reading of a. 
graduated arc can always be carried much lower than the 
divisions actually marked on it. This is sometimes accom- 
plished by the vernier, and sometimes by the reading micro- 
scope. 

49. TJie vernier. — This contrivance, so named from the in- 
ventor, is a short graduated arc, which slides along the limb of 
the circle that is to be subdivided. For example, AB (Fig. 15)^ 
is a vernier for dividing the 12' spaces of the arc on its right 
into portions of 1' each. For this purpose, the vernier consists 
of 12 parts, which together are equal to 11 of the divisions of 
the limb. Since 12 parts of the vernier are 
less than 12 divisions of the arc by a whole 
division, one part of the vernier is less than one 
division of the arc by yV of a division ; two are 
less than two by ^ °f a division, and so on. 
Now, in the figure, the zero of the vernier has 
passed 10° 24/ ; and in order to find how many 
twelfths of the next space it has passed, it is 
only necessary to look along the vernier, and 
observe the number of the division line, which 
coincides with a line of the arc. In this case 
we find it to be the 8th. Hence, the 8 parts 
of the vernier from to 8 are less than the cor- 
responding S divisions of the arc by T 8 2 ; that 
is, zero is T 8 2 of 12' beyond 10° 24\ Therefore 
the reading is 10° 32'. 

The vernier is sometimes made, so that a 
given number of parts equals one more, instead 
of one less, than the same number on the limb. 
Rut the principle of making subdivisions is the same. 



Fig. 15. 



13° 



-12° 



■ir 



■io c 



30 TO FIND DECLINATION. 

50. The reading microscope. — This is a compound micro- 
scope, having in the focus of its eye-piece a pair of spider-linea 
intersecting each other, and in the same field of view are the 
magnified divisions of the arc. The intersection of the spider- 
iines is moved laterally from one division line of the arc to 
another by a screw. If the divisions, for example, are equal to 
5' each, then the screw is so made as to move the intersection 
from one line to another by five revolutions, and therefore each 
revolution indicates a motion of V. A circle is attached to the 
axis of the screw, having its circumference divided into 60 
equal parts. As each revolution ■ of the screw can thus be 
divided into 60 equal parts, so each minute of the arc can be 
divided into seconds. 

One of these reading microscopes is represented at A (Fig. 
14) ; and the places of others are marked at B, C, D, E, F, 60° 
from each other. Six are used, instead of one, for the purpose 
of obtaining a more accurate result, by taking a mean of the 
seconds in the several readings. 

51. To find the declination of a heavenly body. — This may 
be done by measuring its meridian altitude. Let the mural 
circle be adjusted in altitude, so that, at the instant when the 
body crosses the vertical wire of the telescope, it is on the 
horizontal wire also. The graduation of the limb shows its 
altitude. The latitude of the observatory being known, the 
elevation of the equator is known ; and the difference between 
the altitude of the body and the elevation of the equator, is the 
decimation sought. In northern latitudes, if the altitude of the 
heavenly body exceeds the elevation of the equator, the differ 
ence is a northern declination ; if it is less, the decimation is 
south. 

Before altitudes can be measured, the horizontal position ol 
the telescope must be determined. This may be done by 
bisecting the angle between the direction of a fixed star, as seen 
at culmination, and its apparent direction, when seen at 
another culmination in a mirror of liquid mercury, called the 
artificial horizon. By a law of optics, the apparent depression 
below the horizon equals the elevation above it, sc chat the 
whole angle equals twice the altitude. 



ALTITUDE AND AZIMUTH INSTEUMENT. 



31 



5 2. The transit circle. — Sometimes the circle of the transit 
ir. '.trument is made of much larger size than is represented in 
Fig. 11, in order that declinations as well as right ascensions 
may be observed by it. This combination of the transit instru- 
ment and mural circle is called the transit circle, and is con- 
sidered by some practical astronomers to possess an advantage 
over the mural circle in the steadiness of its axis. 

53. The altitude and azimuth instrument. — The essential 
parts of this instrument are, a telescope and two graduated 
circles, one vertical, the other horizontal. Fig. 16 presents one 
of its more simple forms. The telescope AB is movable on a 

Fig. 16. 




horizontal axis at the center of the vertical circle dbc, and also 
on a vertical axis, passing through the center of the horizontal 
circle EFGL The levels g and A, placed at right angles to each 
other, show when the circle EFG- is brought to a horizontal 
position by the tripod screws. The tangent screws, d and <?, 
give slow motions, one in a vertical, the other in a horizontal 
plane. If the reading of the vertical circle is taken when the 



32 THE SEXTANT. 

telescope is horizontal and again when it is directed U a star 
the difference of the readings is equal to the altitude of the star 
In a similar manner, if the horizontal circle is read, when the 
telescope is directed to the north, and read again when it ia 
directed to a star, the difference is its azimuth. 

54. The sextant. — This is an instrument for measuring the 
angular distance between two points situated in any plane 
whatever. It is represented in Fig. IT. I and H are two 
small mirrors, and T a small telescope. ID is a movable radius 
or index, carrying the index mirror at the center of motion, I, 



>— 



Fig. 17. 



*8— 




and a vernier at the extremity, D. The horizon glass, H, is 
silvered only on one-half of its surface. When the zero oi 
the vernier coincides with that of the arc at F, the mirrors 
are precisely parallel. If now we direct the telescope to a 
star, it may be seen in the transparent part of the horizon 
glass, and its image in close contact with it, in the silvered part. 
This is owing to the fact, that a heavenly body is so far dis- 
tant, that the rays from it to the two mirrors are sensibly par- 
allel to each other. 



SPHERICAL PROBLEMS. 33 

55. To measure an angle by the sextant. — Let it he required 
to measure the angular distance between the star S and the 
moon M. The telescope being directed to S, and the sextant 
being held so that the plane of reflection shall pass through the 
two objects, turn the index from F toward E, until the image 
of the moon is brought to the star, its nearer limb just touch- 
ing S. Now, according to an optical principle, the angular 
distance between the moon and its image is just twice that be- 
tween the mirrors. Therefore, by reading the vernier at D, we 
obtain the angular distance between the star and the moon's 
nearer limb. Again, bring the further limb to the star, and 
find its distance. Half their sum is the angular distance be- 
tween the moon's center and the star. 

In like manner, the altitude of a body may be found, by 
bringing its image to coincide w r ith the image of the same body 
seen in the artificial horizon. One-half the angle read from 
the vernier is the altitude of the body. 

The graduation on the limb of the sextant, for convenience, 
corresponds, not to the actual length of the arc passed over by 
the vernier, but to the angular motion of the body, which is 
twice as rapid. Hence, on the arc of 60°, the graduation 
reaches 120° ; and all angles not greater than this can be 
measured by the instrument. 

The two instruments just described are sometimes conven- 
ient at the observatory, but their chief use is elsewhere. The 
altitude and azimuth instrument is of great value in trigono- 
metrical surveying. The sextant is important for the naviga- 
tor, since a stationary instrument 
cannot be employed at sea. Fi S- 18 - 

56. Spherical problems. — 
I. To compute the sun's right 

ascension, declination, or longi- 
tude, or the obliquity of the eclip- 
tic to the equator, when any two 
of the others are given. 

Let PEP' (Fig. 18) represent 
the solstitial colure, PP' the axis, 
EQ the equator, E'C the ecliptic, 

3 




34 SPHERICAL PROBLEMS. 

and PSP' a secondary of the equator passing through the sun 
S. Then SAR is the obliquity of the ecliptic, and RS the dec- 
lination of the sun. And if its longitude is less than 90°, AS 
is its longitude, and AR its right ascension. If its longitude 
is more than 90°, AS and AR are the supplements of longitude 
and right ascension. In both cases the declination is north. 
When the sun's place is represented by S', and its longitude is 
between 180° and 270°, then the longitude = 180° + AS', and 
the right ascension — 180° -f- AR'. But if its longitude is 
more than 270°, longitude — 360° — AS', and right ascen- 
sion = 360° — AR'. In each case the declination is south. 

The triangle ARS is right-angled at R ; and by Napier's 
rule, any one of the parts may be found, whea two others are 
given. 

Ex. 1. When the sun's right ascension is 53° 38', and its dec- 
lination, 19° 15' 57", required its longitude, and the obliquity 
of the ecliptic. 

1. Rad . cos AS =cos AR . cos RS. 

2. Rad . sin AR — tan RS . cot A. 

Ans. Long. = 55° 57' 43". Obi. = 23° 27' 501". 

Ex. 2. On March 31st, the sun's declination was observed to 
be 4° 13' 31|", and the obliquity was 23° 27' 51" ; required 
the sun's right ascension. Ans. 9° 47' 59". 

Ex. 3. What is the sun's longitude in November, when its 
declination is 21° 16' 4", and its right ascension is 16h. 14m. 
58.4s. % Ans. 245° 39' 10". 

The above data show that the sun's longitude is more than 
180° and less than 270°, and the declination south. The tri- 
angle for computation is AR'S'. 

Ex. 4. The sun's longitude being 8 s 7° 40' 56", and the 
obliquity 23° 27' 42£" ; required right ascension in time. 

Ans. lt)h. 23m. 34s. 

II. Given the latitude of a place, and the declination of the 
sun, to find the time of its rising and setting. 

Let PEP' (Fig. 19) be the meridian of the place, Z its zenith, 
and HO its horizon. Let LL' be the diurnal circle of the sun ; 
RS is its declination, S the place of its rising and setting, and 
LS the arc described between either and midnight. But LS, 
in degrees, equals QR, the complement of AR. The angle 



SPHEEICAL PROBLEMS. 



35 



Fig 19. 




S AE = E AH, which is measured by 
EH, the co-latitude, and E is a right 
angle. Therefore, rad . sin AE = 
cot A . tan ES. 

Ex. 1. Eequired the time of sun- 
rise at latitude 52° 13' 1ST., when the 
sun's declination is 23° 28' K 

We find AE=34° 3' 21*"; .\ 
QE = 55° 56' 38f /, = (in time) 3h. 
43m. 46±s. This is the time of sun- 
rise. The same subtracted from 
12h., gives 8h. 16m. 13Js. for the time of sunset. 

Ex. 2. Eequired the time of sunrise at latitude 57° 
K, when the sun's declination is 23° 28' K 

Arts. 3h. 11m, 

Ex. 3. How long is the sun above the horizon in latitude 58° 
12' K, when its declination is 18° 40' S. ? 

Ana. 7h. 35m. 52s. 

In a similar manner, if the declination of any heavenly body 
be given, the interval of time between its culmination, and its 
rising or setting, can be computed. 

III. Given the latitude of a place, and the declination of a 
heavenly body, to compute its altitude and azimuth, when on 
the six o'clock hour-circle. 

Let PEP 7 (Fig. 20) be the meridian of the place, and P the 
elevated pole. Then PP r rep- 



2' 54 



49s. 



resents the six o'clock hour- 
circle, which is at right angles 
to the meridian, and therefore 
projected in a straight line. 
Let the body cross it at S, and 
let ZSB be the vertical circle 
passing through it. In the tri- 
angle ASB, AS is the declina- 
tion, SB the altitude, AB the 
amplitude or complement to 
the azimuth OB, and B is a 
right angle. 
Ex. 1. What were the altitude 



Fig. 20. 




and azimuth of Arcturus, 



36 



SPHERICAL PROBLEMS. 



when on the six o'clock hour-circle, latitude 51° 28' 40" N.", its 
declination being 20° 6' 50" N. I 

^n*. Altitude 15° 36' 27" ; Azimuth 77° 9' 4". 

Ex. 2. In latitude 62° 12' 1ST. the altitude of the sun at sis 
o'clock, a. m., was observed to be 18° 20' 23 // . Eequired its 
declination and azimuth. 

Am. Declination 20° 50' 12" 1ST. ; Azimuth 79° 56' 4". 

LV. Given the latitude of a place and the sun's declination. 
to find the time a. m. when it will cease shining on the north 
side of a building, or the time p. m. when it will begin to shine 
upon it. 

Let PEP 7 (Fig. 21) be the meridian of the place, ZAK the 
prime vertical, and S the place where 
the sun crosses it, and thus ceases to 
shine on the north side of a vertical 
wall. Let PSB be the hour-circle 
through the sun at S. BS is the sun's 
declination, BAS (=EZ) is the lati- 
tude, and AB, changed into time, will 
show how long after six o'clock a. m., 
or before six p. m., the sun transits 
the prime vertical. 

Ex. 1. In latitude 42° 22' 17" K, 
when the sun's declination is 23° 27' 36" K., at what times does 
the sunshine begin and end on the north and south sides of a 
building? Ans. 7h. 53m. 3Ss. a. m., and 4h. 6m. 22s. p. m. 

Ex. 2. How long does the sun shine on the south side of a 
vertical wall, in latitude 20° 30' N., when the sun's declination 
is 20° N? Ans. lh. 45m. 48s. 

Y. The latitude and the sun's declination being given, to 
find the time of day by the sun's altitude. 

Let Z (Fig. 22) be the zenith of the place, P the pole, and 8 
the place of the sun. Measure ZS, the 
zenith distance of the sun, and correct 
it for refraction and parallax. PZ is 
the co-latitude of the place, and PS the 
co-declination of the sun. Therefore, 
the sides of the spherical triangle PZS 
are all known, and the angle ZPS can 





SPHERICAL PROBLEMS. 



37 



Fiar. 23. 




be computed ; which, changed to time, shows how long before 
or after noon the observation was made. 

VI. Given the latitude and the sun's declination, to find the 
time when twilight begins and ends. 

The twilight begins or ends when the sun is about 18° 
below the horizon (Art. 38). Let Z (Fig. 23) be the zenith, P 
the pole, and S the place of the sun at the beginning or ei.d of 
twilight. ZS = 108°, ZP = co-lat, 
PS = co-decl. The three sides of 
ZPS are given, to find the hour-angle 
ZPS. This may be done by dropping 
the perpendicular arc P/>, and using 
the proportion (Sph. Trig.) tan J ZS : 
tan \ (PS +ZP) : : tan £ (PS - ZP) : 
tan \ (Bp — Zp). Having obtained 
Zp and Sp, compute the angles at P, 
and add them together. 

Ex. In lat. 42° 22', when does twi- 
light begin and end, at midsummer, 
the sun's declination being 23° 2S / ? 

Ans. 2h. 6m. 20s. a. m. 

VII. Given the right ascension and declination of a body, to 
find its longitude and latitude. 

Let EQ (Fig. 24) be the equator, and P its north pole, E'O 
the ecliptic, and B. its pole, and 
S the place of the body. Join 
PS and ES, and draw the arc 
SB perpendicular to PC. PS, 
the complement of declination, 
is known ; likewise RP, which 
equals EE', the obliquity. As 
A is the vernal equinox, SPQ 
is the complement of right as- 
cension, and therefore known. 
SRC is the complement of lon- 
gitude, and PS is the comple- 
ment of latitude. 

In the right-angled triangle PSB, PS and P being known, 
find PB. Then KB(=RP+PB) is known. Then (Sph. 



9h. 53m. 40s. p. m. 



Fig. 24. 




38 THE sun's eight ascension. 

Trig.) sin KB : sin PB : : tan P : tan E. Thus R, the com 
plement of longitude, is found. Then, in the right-angled 
triangle BSB, BB and the angle K enable us to find BS, the 
complement of latitude. 

Ex. 1. The right ascension of a planet was observed to be 
82° 7', and its declination 24° 26' K Calling the obliquity 
23° 27' 20", what were the longitude and latitude of the 
planet ? Ans. Long. 82° 49' 30" ; Lat. 1° 10' 27" K 

Ex. 2. What are the longitude and latitude of the star, 
whose right ascension is 4h. 40m. 49s., and its declination 66* 
6' 37" K % Ans. Long. 79° V 8" ; Lat. 43° 24' 5" K 



CHAPTEB IV. 



THE EARTH S ANNUAL MOTION ABOUT THE SUN. — THE 
SEASONS. — FIGURE OF THE EARTH'S ORBIT. 

57. Ohservatums of the suns place. — If we employ the in- 
struments of the observatory in measuring from day to day the 
right ascension and declination of the sun, at the moment of its 
crossing the meridian, it will be discovered that these quantities 
are constantly changing ; or, in other words, that the sun is 
constantly shifting its place in relation to the stars. 

58. Its right ascension. — By the transit instrument and 
clock, it is found that the sun's right ascension is always in- 
creasing by a quantity which is not quite uniform, but which 
amounts to nearly one degree every day. So that, in about 
36 5 days, it describes the whole 360° of right ascension, and 
appears again in the same place among the stars. This is the 
apparent annual motion of the sun, by which it seems to pass 
round the heavens from west to east once in a year. 



THE TROPICS AND POLAR CIRCLES. 39 

59. Its declination. — But while thus passing round, it also 
moves alternately north and south. For, by measuring the 
declination each day by the mural circle, it is found that after 
passing the vernal equinox, March 20th, its declination is 
north, and increases to the summer solstice, June 21st, when it 
reaches nearly 23^° ; from that point it diminishes to zero at 
the autumnal equinox, September 22d. The declination then 
becomes south, increasing to the winter solstice, December 21st, 
when it is 23J°, and thence diminishing to nothing at the 
vernal equinox, on March 20th of the following year. 

60. The ecliptic. — The apparent annual path of the sun is 
found by the foregoing observations to lie in a plane, cutting 
the celestial sphere in a circle called the ecliptic (Art. 12), and 
inclined to the plane of the equator at an angle of about 23° 27'. 
This plane maintains almost a constant position among the 
stars, and is used far more than any other circle of the sphere 
as a plane of reference. 

The obliquity of the equator to the ecliptic in 1850 was 
23° 27' 31", and diminishes at the rate of 46" in a century. 

6 1 . The zodiac. — This name is given to a zone of the heav- 
ens, 16° wide, extending along the circle of the ecliptic, 8° on 
each side of it. The paths of the principal planets lie within 
this zone. Its length is divided into 12 signs of 30° each, 
having the same names and arranged in the same order as 
those of the ecliptic (Art. 13), though not coincident with them. 
The signs of the zodiac are distinguished from each other by 
the stars which occupy them. 

62. The tropics and polar circles. — Through the two points 
of the ecliptic most distant from the equator, called the sol- 
stices (Art. 12), we imagine circles to be drawn parallel to 
the equator, called the tropics. The northern circle, passing 
through the first of Cancer on the ecliptic, is called the tropic 
of Cancer ; the southern one, for a like reason, is called the 
tropic of Capricorn. Two other parallels to the equator, passing 
through the poles of the ecliptic, and therefore 23 d 27' from the 
.poles of the equator, are called the polar circles. 



40 



ANNUAL MOTION. 



63. Terrestrial zones. — On the terrestrial sphere, a similar 
system of circles divides the earth's surface into the well-known 
zones of geography, called the torrid, temperate, and frigid 
zones. The tropics are the limits of vertical sunshine in mid- 
summer. The polar circles are the limits within which the 
sun makes a diurnal revolution in midsummer and mid-winter, 
without rising or setting 

64. The annual motion observed without instruments. — If 
the stars were visible in the daytime, we should perceive the 
sun making progress among them toward the east, by a dis- 
tance equal to nearly twice its own breadth every day, since 
the apparent diameter of the sun is a little more than half a 
degree. But, as they are invisible by day, we detect the same 
fact, when we notice that at a given hour of the night, all the 
stars are further west than on a previous night. For example, at 
9 o'clock p. m. — that is, 9 hours after noon — it is easily observed 
that there is, from one evening to another, a regular progress 
of all the stars westward, as long as we choose to watch them. 
In other words, the sun is at the same rate advancing eastward 
relatively to the stars. 

Fig. 25. 




65. The annual motion is a motion of the earth, not of the 
mn. —There is abundant evidence that the motion of the sun 



CHANGE OF SEASONS. 



41 



around the earth, above des3ribed, is only apparent, and 
results from a real motion of the earth about the sun. Thus, 
suppose the earth to pass around the sun S (Fig. 25) in the 
orbit ABPC, in the order of the signs ; if we were unconscious 
of this motion, the sun would appear to us to move about the 
earth in the same order of the signs, though, at any given 
moment, in a contrary direction. When the earth is at B (in 
the sign T, as seen from the sun), we should see the sun in the 
sign =a= ; when we reach tf , the sun is seen in ^l ; and so on. 

66. Cause of the change of seasons. — The changes of the 
seasons are due to the fact, that the two revolutions of the 
earth, one on its axis, and the other around the sun, are in dif- 
ferent planes ; in other words, that the equator and the ecliptic 
make an angle with each other. In Fig. 26, let ABCD repre- 




sent the ecliptic (seen obliquely), and suppose the earth to 
pass around in the order of the letters, occupying the position 
A on the 20th of March, B on June 21st, C on Sept. 22d, and 
D on Dec. 21st. In every position of the earth, the equator eq, 
is inclined the same way, and always at the angle of 23^° with 
the ecliptic. The axis ns, being perpendicular to the equator, 
is everywhere parallel to itself. 

When the earth is at A , the vernal equinox, the line of inter- 
section of the ecliptic and equator, passes through the sun S, 
and the light just reaches the poles n and s ; so that, as the 
earth rotates on ns, every place is one-half of the time in the 
light, and the other half in darkness. The days and nights are 
therefore equal. 

As the earth passes on toward B, the light readies beyond 



42 



HEAT IN SUMMER AND COLD IN WINTER. 



the north pole more and more, till at B, the summer solstice, 
it extends 23-J- beyond n, and falls as much short of s, the sun 
being now north of the equator eq. As the earth now rotates 
on ns, all places north of eq are in the light longer than in the 
shade, and the reverse is true of all places south of eq. It is 
summer in the northern hemisphere, and winter in the southern. 

On the 22d of September, the earth arrives at C, the autum- 
nal equinox ; the intersection of the two planes again passes 
through the sun, the light once more reaches the poles, and the 
days and nights are equal. 

At D, the winter solstice, the north pole n is turned as far 
as possible into the shade, and s into the light. Every place 
north of eq is in the light a shorter time than in the darkness, 
and the reverse south of eq. It is now winter in the northern 
hemisphere, and summer in the southern. 




If the equator were in the same plane with the ecliptic, the 
case would be represented by Fig. 26a. The axis ns would 
then be perpendicular to the ecliptic as well as to the equa- 
tor, the circle of illumination would always reach just to the 
poles n and s, and in the daily rotation, every place would be 
half the time in the sunlight, and half in the darkness. There 
would, therefore, be no inequality of day and night, and no 
change of seasons. 

67. Causes of heat in summer and cold in winter. — These 
are two. 

1st. The length of the aay compared with the night. The 
heat of the earth is passing off by radiation during the whole 
time, whether the sun shines or not But the earth receives 







GREATEST HEAT AND GREATEST COLD. 43 

beat from tlie sun, only while the sun is above tho horizon. 
Hence, the longer the period of sunshine, compared with the 
time of a diurnal revolution, the greater the heat. For this 
reason, therefore, the summer is warmer than the winter. 

2d. The different inclination of the rays to the general sur 
face of the earth. The number of rays falling on a given sur- 
face, varies as the sine of inclination. Let AB (Fig. 27) be 
the breadth of the surface. If the rays fall on it at the 
angle ABC, the perpendicular 
breadth of the beam is AC ; if 
at the angle ABD, the breadth 
of the beam is AD ; while, if 
they fall perpendicularly, the 
breadth of the beam is AB 
itself. Now, the number of 
rays in the beam obviously 
varies as its perpendicular 
breadth. But these breadths, 
AC, AD, and AB, are as the sines of the several inclinations. 
In summer, the sun rises to a greater elevation each day than at 
other seasons, and therefore sheds a greater quantity of heat on 
that part of the earth. 

Ex. 1. What is the relative quantity of direct heat from the 
Bun at noon, on two equal horizontal areas, one in latitude 75° 
N., the other 30° K, when the sun's declination is 19° 1ST. ? 

Ans. As 100 : 175J. 

Ex. 2. Find the ratio, as in Ex. 1, in latitude 50° !N". and 
latitude 45° S., when the sun's declination is 15° 45' S. 

Ans. As 100 : 212.4. 

68. Why the greatest heat is later than the summer solstice, 
and the greatest cold later than the winter solstice. — If the sun 
sheds on a given surface more heat each day than the surface 
loses by radiation, then the heat accumulates from day to day. 
This is the case during the long days of summer ; and more 
heat is gained than lost, till a month or more after the summer 
solstice. For a like reason, during the middle hours of the day, 
heat is received from the sun more rapidly than it is lost by 
radiation, so that the hottest hour is 2 or 3 o'clock p. m. 



44 GREATEST CHANGES OF SEASON. 

In the winter, on the contrary, the loss by radiation exceeds 
the quantity received from the sun, during all the shortest days, 
so that the temperature descends till many weeks after the 
winter solstice. 

If loss by radiation were at a uniform rate at all tempera- 
tures, and the temperature of successive years should remain 
constant, as it now is, then the greatest heat would be near the 
autumnal equinox, and the greatest cold near the vernal equi- 
nox, the times when the surface receives heat at the mean rate. 

On the contrary, if the existing amount of loss by radiation 
were distributed so as to be exactly proportional to the acces- 
sions received from the sun, there would be no change of tem- 
perature at the different seasons of the year or the different 
hours of the day. 

But the radiation of heat follows neither of these laws ; the 
quantity radiated is greater, when the quantity received is 
greater, but it does not vary at so rapid a rate. 

69. No change of seasons, if there were no obliquity. — The 
angle between the planes of the two motions of the earth being 
the cause of the change of seasons, it follows that there would 
be no such change if those motions were in the same plane. If, 
while the earth advances in its orbit about the sun, it should 
rotate in the same direction on its axis, then the sun would 
always be in the plane of the equator, and would, every day, 
describe the equator as its diurnal circle, rising exactly in the 
east, culminating at a zenith distance equal to the latitude of the 
place, and setting exactly in the west. At the equator, the sun 
w r ould always follow the prime vertical, and at either pole it 
would always be passing round in the horizon. See Fig. 26a. 

7 0. The greatest changes of season, if the obliquity were 
90°. — If, while the earth revolves on its axis from west to east, 
it should pass around the sun in a plane lying north and south, 
then the ecliptic would pass through the north and south poles, 
and the solstices would be at the poles. Hence, at a station on 
the equator, the sunw T ould, during the year, describe the prime 
vertical and various small circles parallel to it, down to the 
oorth and south points of the horizon, where it would be 



FORM OF THE EARTH S ORBIT. 



45 



stationary alternately at the times of the solstices. At the 
equator, therefore, there would be an alternation from summer 
to winter, or the reverse, every three months. 

At either pole there would be but one summer and one 
winter in a year; but the extremes would be far more intense. 
For the sim, in describing diurnal circles parallel to the 
horizon, would occupy six months in ascending to the zenith 
and returning to the horizon, and the remaining six months in 
performing corresponding revolutions below the horizon. 

At intermediate places, the extremes of the seasons would 
also be intermediate. 



7 1 . Mode of determining the form of the earth? s orbit. — 
The earth's orbit is an ellipse described about the sun, which is 
situated in one of its foci. This is ascertained by observing the 
changes in the sun's apparent diameter throughout the year^ 
When the sun appears smallest, it is most distant ; and when 
largest, it is nearest. And its distance, in all cases, varies in- 
versely as its apparent diameter. Therefore, if the sun's angu- 
lar diameter be accurately measured as frequently as possible, 
the reciprocals of those angles express the relative distances ; 
and these distances determine the form of the orbit. 

Thus, suppose the earth to be at E (Fig. 28), and that the 
sun's apparent diameter is 



measured when in the di- 
rection E#. After it has 
advanced eastward some 
days, so as to be seen in the 
direction E&, let another 
measurement be made ; and 
so on, at every opportunity 
through the year. Then 
let E«, E5, Ec, etc., be made 
proportional to the recipro- 
cals of the apparent diame- 
ters, and be laid down at 
angles equal to the angular 
changes of the sun's place. 



Fig. 28. 




A line, a b m v, passing through 



their extremities, shows the form of the sun's apparent orbit 



46 LINE OF APSIDES, 

about the earth, and therefore the form of the earth's real orbit 
about the sun. 

In this manner, even while ignorant of the size of the orbit, 
we learn that its form is an ellipse, and that the sun occupies 
one of its foci. 

72. Definitions relating to a planetary orbit. — Let E be the 
focus occupied by the sun, and am the major axis of an ellipti- 
cal orbit described about it ; the nearest point, a, is called the 
perihelion, and the most distant point, m, the aphelion. The 
two p©ints a and m are also called the apsides. The point a 
is sometimes called the lower apsis, and m the higher apsis. 
The varying distance, E&, E5, E^, etc., is called the radius 
vector. If the major axis, am, is bisected in C, the ratio of EC 
to the semi-major axis, aC, is called the eccentricity of the 
orbit. The less EC is, compared with aC, the less is the 
eccentricity, and the nearer does the ellipse approach to a 
circle. If E coincides with C, the eccentricity is nothing, and 
the orbit is a circle. 

73. The earthus orbit very nearly circular. — The eccen- 
tricity of the earth's orbit in 1850 was 0.01677, and is very 
slowly diminishing. This fraction is about -g^, — that is, EC 
(Fig. 28) is eV of aC. As aC, in this figure, is about one inch 
long, EC should be only ^ of an inch, in order to represent 
correctly the proportions of the earth's orbit. If it were thus 
drawn, it could not be distinguished from a circle in its appear- 
ance ; for the minor axis, as may be easily computed, would be 
shorter than the major axis by only to ff o o of an inch. 

74. Position of the line of apsides. — The direction of the 
major axis of the earth's orbit, or the line of apsides, is slowly 
changing ; but at present it passes through the 1 0th degree of 
Cancer and Capricorn, as represented in Fig. 25. The earth 
is at perihelion on the 1 st of January, and at aphelion on the 
1st of July. We are therefore nearest to the sun in the winter 
of the northern hemisphere, and furthest from it in the summer. 

7 5 , Distance from the sun, as affecting the seasons. — The 






SIDEREAL TIME. 47 

intensity of the sun's heat at the earth, as well as that of its 
light, varies inversely as the square of our distance from it. 
On this account, the intensity of heat at perihelion is to that at 
aphelion as 61 2 : 59 2 , which is nearly as 31 : 29. Therefore, 
so far as distance is concerned, the earth receives ^ more heat 
on the 1st of January than on the 1st of July. This produces 
a slight effect to mitigate the severity of cold in winter and o* 
heat in summer, in the northern hemisphere, and to aggravate 
the same in the southern hemisphere. But, on account oi 
changes going on in the places of the equinoxes and apsides, 
this modifying effect will be reversed after the lapse of about 
10,000 years. 



CHAPTEK Y. 



SIDEREAL TIME. — MEAN AND APPARENT SOLAR TIME. — 
THE CALENDAR. 

76. The sidereal day. — This is the interval of time which 
elapses between two successive culminations of a star (Art. 43). 
The length of this interval appears to be invariable, whatever 
star is observed, or in whatever season or year the observation 
is made. On this account, the sidereal day is regarded as the 
true period of the earth's rotation on its axis. In order to 
reckon by sidereal time, the moment chosen for the beginning 
of each sidereal day is the moment when the vernal equinox 
culminates. The sidereal clock, if correct, then points to 
Oh. 0m. 0s. Each sidereal day is divided into 24 sidereal hours, 
each hour into 60 sidereal minutes, and each minute into 60 
sidereal seconds. 

7 7 . The w.ean solar day.- -This is the mean interval between 
two successive culminations of the sun. It will be shown pres- 
ently, that these intervals vary throughout the year. As the 
sun, by the annual motion, is advancing eastward continually 
among the stars, the solar day must always be longer than the 



48 INEQUALITY OF SOLAR DATS. 

sidereal day. For, if the sun and a star were 01 the rnendiafi 
of a place together, then, while that place passes around east- 
ward till its meridian meets the star again, the sun has ad- 
vanced eastward nearly a degree, and the place must revolve 
nearly a degree more than one revolution before its meridian 
will reach the sun. This will require nearly 4 minutes of 
time; for, in the diurnal motion, 15° correspond to one hour, 
and therefore 1° to y 1 - of an hour — that is, four minutes. 

78. The relation of sidereal time to mean solar time. — As 
the sun, in its apparent annual motion, describes 360° in 
365.24 days, it will, in one day, on an average, pass over 360° 
-7- 365.24 = 59' 8.35", or nearly 1°, as before stated. But, by 
the diurnal motion, a given place on the earth in one solar day 
describes 360° plus the above arc. Therefore, 360° 59' 8.35" : 
59' 8.35" : : 24h. : 3m. 55.9s. of solar time. This is the excess 
of the mean solar day above a sidereal day. And one sidereal 
hour, minute, or second is to one solar hour, minute, or second 
as 360° : 360° 59' 8.35",— that is, as 1 : 1.0027379. Therefore, 
to reduce a given period of time from the mean solar to the 
sidereal reckoning, multiply by 1.0027379 ; and to reduce side- 
real time to mean solar time, divide by the same number. 

79. The apparent solar day. — This is the actual interval 
between two successive culminations of the sun. And this in- 
terval changes its length from day to day through the entire 
year, being sometimes greater, and sometimes less than the 
mean solar day. 

In keeping solar time by clocks and watches, it is customary, 
for convenience, to aim to keep the mean rather than the 
apparent time, and to regard the sun as going alternately too 
fast and too slow. 

80. First cause of inequality in apparent solar days. — One* 
cause of inequality of days, as measured by the sun, is found 
in the elliptical form of the earth's orbit, and the consequent 
unequal increments of longitude made by the sun from day to 
day. At P (Fig. 25) the sun is nearest to us, and at A it is 
most distant. The motion in the parts of the orbit near P 



INEQUALITY OF SOLAR DATS. 49 

would therefore appear greater than in the parts near A, even 
if it were uniform in all parts. But, besides this, as will be 
shown in Chapter Till , the motion is really greatest at P and 
least at A. For both these reasons, then, the sun, while in the 
nearer half of its orbit, passes over the longest arcs each day 
in the ecliptic — that is to say, in longitude — and the shortest 
arcs, in the half most distant from us. The sun, in fact, occu- 
pies nearly 8 days more time in describing the remote hah 
than the nearer one. 

Eecollecting, now, that a solar day consists of a sidereal day, 
plus the time of describing diurnally the arc which the sun, in 
the mean time, advances annually, it is clear that if this daily 
arc is longer, the solar day is longer ; and if shorter, the solar 
day is shorter. 

So far as this cause is concerned, therefore, the longest solar 
day would be the 1st of January, and the shortest, the 1st of 
July ; and about half-way from P to A, and from A to P, the 
apparent days would have their mean length. 

81. Second cause of inequality in apparent solar days. — 
But the solar days are unequal for another reason — the ob- 
liquity of the ecliptic to the equator. Time is measured by arcs 
of the equator. But the sun's daily advance toward the east 
is made in the ecliptic. Even if the daily increments of the 
sun's longitude were equal, those of its right ascension would 
be unequal, and therefore the solar days unequal. 

Let Fig. 29 repre&ent a portion of the celestial sphere, AF 
a part of the equator projected in a straight line, OH a cor- 
responding part of the ecliptic, Q the vernal equinox, S the 
summer solstice, and P the north pole. Draw through P a 
few meridians, dividing that part of the ecliptic near Q into 
short arcs, to represent the daily increments of the sun's longi- 
tude on CH, and of its right ascension on AF. These meri- 
dians are oblique to CD, but perpendicular to AB. Hence, as 
AQC is a right-angled triangle, QC is longer than AQ ; so also, 
DQ is longer than BQ ; and thus each part of CD is longer 
than the corresponding part of AB ; — that is, the increment! 
of the sun's right ascension, near the equinoxes, are less than 
those of its longitude. The obliquity, therefore, by short- 

4 



50 INEQUALITY OF SOLAR DAYS. 

ening these increments of right ascension, shortens the soja? 
days. 

Bat if meridians are drawn to that part of the ecliptic near S, 
the arcs GH and EF are abont parallel to each other, and the 
increments on the equator are not shortened, as they are at Q 
But, on the other hand, the divergency of the meridians causes 
EF to be longer than GH, and each part of EF longer than 
the corresponding part of GH. At the solstices, therefore, the 
increments of right ascension are lengthened by the divergency 
of the meridians, and hence the solar days are lengthened also. 
About midway between the equinox and solstice, the two 
effects just described neutralize each other, and the daily arcs 
of right ascension, so far as this cause is concerned, are at their 
mean value. 

Fig. 29. 




82. Location of extreme and mean solar days from each 
cause. — Suppose the first cause alone in operation, and that the 
sun and a uniform clock agree with each other at P (Fig. 25), 
on the 1st of January. Then, as the solar days are longer than 
their mean, the sun becomes slower, compared with the clock, 
from day to day, for about three months, when the days will 
have reached their mean length, at a point near half-way from 
P to A. Afterward, the days being diminished below the 
mean, the sun slowly gains on the clock, and catches up with it 
at A, July 1st. But the days now being shortest of all, the sun 
is immediately in advance of the clock, and most of all at a 
point half-way frcm A to P. The gain and loss compensate 



EQUATION OF TIME. 52 

each other from A to P, as they did from P to A. Thus mean 
and apparent time would agree twice in a year, at intervals of 
six months, if eccentricity of orbit were the only cause of 
irregularity. 

Again, if the second cause alone existed, and we suppose the 
sun and clock to agree at the equinox Q (Fig. 29), then the sun 
gains on the clock every day, on account of the short arcs of 
right ascension near Q. In about 1| months, however, the 
days reach their mean length, the sun begins to lose what it 
has gained, and at S, June 21st, the sun and clock are again to- 
gether. But the sun is now losing, falls behind the clock, and 
is furthest behind midway between the solstice and the next 
equinox. The autumnal equinox and the winter solstice are, in 
like manner, points of time at which the clock and sun agree 
with each other. Thus, if the second were the only cause of 
^regularity, the mean and apparent time would agree four 
times in a year, at intervals of about three months each. 

83. The equation of time. — The difference between mean 
time and apparent time, on any given day, is the equation of 
time for that day. If the sun is slow, the equation must be 
added to the apparent time ; if fast, it must be subtracted from 
it, in order to give mean time. 

We have seen by the two preceding articles that, on account 
of eccentricity of orbit, the equation would be reduced to zero 
twice in a year ; and, on account of obliquity of ecliptic and 
equator, it would be zero four times in a year. The joint effect 
of these two causes is, to reduce the equation to zero four times 
in a year, at unequal intervals of time. 

84. The equation of time represented graphically. — The 
ordinates of the curves in Fig. 30 exhibit to the eye the equation 
of time as depending on each cause by itself, and on the two 
conjointly. The relative lengths of the ordinates above and 
below AB show the positive and negative equations, as caused 
by eccentricity, and those on CD the equations as caused by 
obliquity ; while the algebraic sum of these on each vertical 
line, gives the resultant effect on the line EF. The figure 
shows that the equation reaches its first maximum, + 14 
minutes, on the 11th of February; its first minimum, — 4 



52 



THE JULIAN CALENDAR. 



minutes, May 14th ; its second maximum, + 6 minutes, July 
2 6 tli ; and its second minimum, — 16 minutes, November 2d. 
The four times of agreement, when the equation is zero, are 
shown by the intersections ; they occur April loth, June 15th, 
September 1st, and December 24th. The sign + shows that the sun 
is on the meridian after mean noon, the sign — before mean noon. 





Jan. 


Feb. 


Mar. 


April. 


May. 


Fig 

June. 


. 30. 

July. 


Aug. 


Sept 


Oct 


Ncv. 


Dec. | 




























A 


























-"" 




C 






+ 


14 










+ 6 






















^-4^ 












16 





85. Civil and astro?wmical time. — The mean solar day, when 
employed for civil purposes, is supposed to begin and end at mid- 
night, and is divided into hours, numbering from 1 to 12 a. m., 
and then from 1 to 12 p. m. But the astronomical day (which is 
also the mean solar day) begins and ends at noon, 12 hours 
later than the corresponding civil day, and its hours are counted 
from 1 to 24. Thus, the astronomical date, April 12d. 20h., is 
the same as the civil date, April 13th, 8 o'clock a. m. 

S6. The Julian calendar. — The period in which the sun 
passes from the vernal equinox to the same point again, is 
called the tropical year. In that period the round of the 
seasons is exactly completed. The length of the tropical year 
is 365d. 5h. 48m. 46.15s. This is so near 365i days, that in 
the adjustment of the calendar made by Julius Caesar (hence 
called the Julian calendar), three successive years were made 
to contain 365 days each, and the fourth 366 days. The addi- 
tional day is called the intercalary day. In this calendar it 
was introduced by reckoning twice the 6th day before the 



THE GKEGOKIAN CALENDAR. 53 

Kalends of March ; and hence the year containing this addi- 
tional day was called the bissextile. The intercalary day is 
now the 29th of February , and the year containing such a day 
is called leap-year, 

87. The Gregorian calendar. — By calling the tropical year 
865^ days, the Julian calendar makes it more than 11 minutes 
too long, and the intercalation of one day in four years is 
therefore too great. This excess amounts to more than 18 
hours in a century. Hence, by dropping the intercalary day 
three times in four centuries, the adjustment is nearly complete. 
The Julian calendar, thus amended, is called the Gregorian 
calendar, because adopted under Pope Gregory XIII. At that 
time, 1582, the vernal equinox, by the error of the Julian cal- 
endar, had fallen back to March 11th. To bring the equinox 
to its proper date, 10 days were first dropped (the 5th being 
•called the 15th), and then the following system was adopted. 

Every year, not exactly divisible by 4, has 365 days. 

Every year, divisible by 4, and not by 100, has 366 days. 

Every year, divisible by 100, and not by 400, has 365 days. 

Every year, divisible by 400, has 366 days. 

The Gregorian calendar will not be correct perpetually, but 
the error will not amount to a day in 4,000 years. 

The nation of Eussia has not yet adopted the Gregorian cal- 
endar, so that there is now a discrepancy of 12 days between 
their dates and those of other nations. The reckoning still 
used by them is known as old style, and is distinguished by 
appending the letters O. S. to every date. 

88. How to compare days of the month and of the week in 
passing from one year to another. — A common year of 365 
days contains 52 weeks and one day ; a leap-year contains 52 
weeks and two days. Hence, a year usually begins a day later 
in the week than the year previous. And, generally, any day 
of any month is one day later in the week than the same day of 
the preceding year. Thus, July 4th, 1884, falls on Friday; 
1885, on Saturday; 1886, on Sunday. But, in leap year, 
this rule applies only till the end of February. From that 
time to the same date in the year following, every day of a 



54 CENTRIFUGAL FORCE. 

month falls two days later in the week than in the previous 
year. Thus, July 4th, 1883, is Wednesday ; 1884, Friday. And 
February 2d, 1884, is Saturday; 1885, it is Monday. 

Table I., at the end of the volume, contains a complete cal- 
endar for 77 centuries. 



CHAPTEE YI. 



CURVILINEAR MOTION. — SPHEROIDAL FORM OF THE EARTH. — 
ITS DENSITY. — PROOFS OF ITS ROTATION ON AN AXIS. 

89. Projectile and centripetal forces. — Motion in a curve 
line is always the effect of two forces ; one, an impuhe which, 
acting alone, would have caused a uniform motion in a straight 
line, and whose influence is always retained in the curve 
motion ; the other, a continued force, which constantly urges 
the moving body toward some point out of the original line ol 
motion. The "first is called the projectile force, the other the 
centripetal force. If the action of the latter were to cease at 
any moment, the body by its inertia would from that moment 
continue uniformly in the direction in which it was then mov- 
ing. Such motion in the tangent may be regarded as the 
effect of an impulse first given in the direction of that tangent. 
This supposed impulse is the projectile force for the moment in 
question ; but it is in truth the resultant of the original im- 
\ ulse, and the infinite series of actions already produced by the 
centripetal force. 

The centripetal force may be resolved into two components 
one in the direction of the tangent, the other perpendicular to 
it. The tangential component will accelerate or retard the 
motion in the curve according as it acts with the projectile 
force or in opposition to it. When the body moves in the cir- 
cumference of a circle, the tangential component of the cen- 
tripetal force is 0, and hence the motion is un form. 

90. Centrifugal force, — When a body moves in a curve, 



CENTRIFUGAL FORCE. 55 

since by its inertia it tends to proceed in the tangent at that 
point, there is a continual outward pressure directed from the 
center of force: this is called the centrifugal force. It is 
always opposed to the centripetal force, and in circular motion 
is always equal to it. It must not be viewed as a third force 
introduced to explain curvilinear motion, but as that compo- 
nent of the projectile force which acts in opposition to the cen- 
tripetal force. 

01. First law of centrifugal force in circular motion. — 
When a body moves in a circular path, its centrifugal (or cen- 
tripetal) force varies as the square of the velocity divided by the 
radius. Let Ab (Fig. 31) = v, the space passed over in one 
second. The projectile force is then represented by AB, and 
the body would move in that line uniformly, 
were it not for the centripetal force acting 
toward E, and thus deflecting it into Ab. Aa 
being the distance through which the body 
falls in one second, 2A# or c represents the 
centripetal force. Let AE = r. Then Aa : 

v 
Ab : : Ab : AD : or ^-c : v : : v : 2r, and c = —. 



As the centripetal and centrifugal forces are 
equal in circular motion, c may represent either in value, though 
they are opposite in direction. Hence, in a given circle, where 
r is constant, the force either toward or from the center varies 
as v 2 } the square of the velocity. In whirling a ball, for in- 
stance, with a string of given length, if the velocity is doubled, 
the strain upon the string (the centrifugal force) is four times 
as great, and the strength of the string (the centripetal force) 
needs also to be four times as great. So, if a train of cars goes 
round a curve with a velocity 1J times that which is intended, 
its tendency to be thrown from the track is increased 2J times. 

93. Second law of centrifugal force in circular motion. — 
When the path of a body is circular, its centripetal or centrif- 
ugal force varies as the radius of the circle divided by the 
square of the time of revolution. 

Let t = the time of describing the whole circumference 2jtr ; 




56 



LOSS OF WEIGHT. 



and let the velocity per second 

%nr 4tt 2 r 2 

t 

r a * 2rr 2 r 

varies as 



Therefore 2nr = i>£, and 



v = 



* 2 



£ 2 
Aa or ^c 



But (Art. 91) c = f = 



4tT 2 7' 



which 



Hence, if the time of revolution is the same, the attraction to 
the center must be increased as the radius is increased ; for then 
coor. Thus, if a string is twice as long, it must have twice 
the strength, in order to whirl a ball at the same rate oi 
revolution. 

93. Centrifugal force on the eartKs surface. — As the earth 
makes its diurnal rotation, all free particles upon it are in- 
fluenced by the centrifugal force. Let NS (Fig 32) be the 
axis, and A a particle describing a circle with the radius A(X 
If AB, in the plane of that 
circle, represent the centrifugal 
force, resolve it into AD on 
CA produced, and AF, tan- 
gent to the meridian 1STQS. 
The effect of AD is to dimin- 
ish the weight of the particle, e 
while the effect of AF is to 
urge it horizontally toward the 
equator. If the surface, then, 
consists of yielding matter, as 
water, the spherical form can 
not be retained, but the parts about the poles, N and S, will be 
depressed, and those about the equator, EQ, will be elevated. 
At each point between the pole and the equator, a particle is. 
held in equilibrium, by that component, AF, of the centrifugal 
force which urges it toward the equator, and that component 
of gravity which urges it down the inclined surface toward the 
pole. 




94. Loss of weight at the equator caused by rotation. — Let 
the weight of a body, w, be taken to express the force ot 
gravity, and let \g (= 16^ feet) be the distance fallen through 
by this body in one second. Kow, c is the force by which Aa 



LOSS OF WEIGHT AT EQUATOR. 57 

2n 2 r 

(Fig 31) is described in one second ; and Aa = — ^— (Art. 92} 
Hence, 



w : c :: 


\g: 


2ttV 


.\ c = 


w X 


j2 « 







Using the values of the letters in the fraction, we obtain c, the 
centrifugal force, in terms of w, the weight of the body. 

The equatorial radius of the earth, r, is 3962.8 miles = 
20,923,584 feet. 

The earth makes one rotation in 24 sidereal hours = 86,400 
sidereal seconds. Reducing this to solar seconds (Art. 78), we 



find 



c = w X 



t = 86,164s. Hence, 

4 x 3.14159 2 x 20,923,58 4 = _w_ 
321 x 86,164 2 ~ 289* 



And, since the centrifugal force at the equator acts directly 
from the center, a body at the equator loses ^ig of its weight 
by the rotation of the earth. 

95. Loss of weight by rotation at other latitudes. — Since c 
varies as r (Art. 92), the centrifugal force is greatest at the 
equator, and zero at the poles, and the force at the equator 
is to that at any latitude A (Fig. 32) as QC : AO — that is, as 
rad : cos lat. But, except at the equator, the centrifugal force 
does not directly oppose gravity. If AB is the whole centrif- 
ugal force at A, AD is the component of it which acts agains* 
gravity. But AB : AD : : AC : AO : : rad : cos lat. So thai 
the loss of weight is diminished again in the same ratio as be- 
fore. Tlierefore, the loss of weight at the equator is to that at 
any given latitude, as rad 2 : cos 2 of latitude. 

96. Whole loss of weight at the equator. — It is found by 
observations made with the pendulum, that the weight of a 
body at the equator is y^- less than that at the poles. But the 



58 



SPHEROIDAL FORM OF THE EARTH, 



loss from centrifugal force is only ^ig. Subtracting this from 
j-jL^, the remainder is very nearly ^ 6 , a loss of weight at the 
equator which must be ascribed to some other cause. This 
cause is the oblateness itself, by which the equator is more 
distant from the center than the poles are. 



97. Spheroidal form of the earth found by measurement — 
Not only is the oblate form of the earth inferred from its rota- 
tion on its axis, but the measurement of the length of a degree 
of latitude, at various distances from the equator, proves that 
the meridians of the earth are ellipses, whose major axes are in 
the plane of the equator, and their common minor axis a line 
joining the poles. If the meridians were circles, all the degrees 
of latitude would be of the same absolute length, but it has 
been ascertained, by numerous and most accurate trigometri- 
cal surveys, that the length of a degree of latitude is least at 
the equator, and increases toward the poles. But if the degree 
lengthens as we go toward the pole, then the radius must 
lengthen in the same proportion, and therefore the curve, belong- 
ing to a larger circle, must become more flattened. And this 
change of curvature belongs to an ellipse, not to a circle. Thus, 
at Q (Fig. 33) the degree is shortest, longer at K, still longer at 
L, and so on to the pole. 
The center of the arc Q 
is at A, nearer than the 
center of the earth ; the 
center of K is B, of L is 
D, and of the polar arc 
it is F, beyond the cen- 
ter C. Thus, the cen- 
ters of curvature of the 
elliptical quadrant Q~N 
lie on the curve ABDF, 
which is the evolute of 
that quadrant. Each 
meridian quadrant is in 
like manner the involute of a curve, and their four evolutea 
form the figure AFG-H about the center. Eo part of a meri- 
dian has its center of curvature at the center of the earth. 




MASS OF THE EAKTH. 



59 



The following numbers express both the size and the form of 
the earth : 



Equatorial diameter 
Polar diameter 
Mean diameter 
Difference of diameters 



7925.604 miles. 
7899.100 " 
7912.357 " 
26.504 " 



The difference of diameters is ^^-g of the equatorial diani 
eter ; this is called the compression of the poles, or the ellip- 
iicity of the earth. 

So slight is the oblateness above described, that an exact 
model of the earth could not be distinguished by sight or touch 
from a perfect sphere. 

The volume of the earth = (7912.357) 3 x g = 259,400,000,000 

cubic miles. 

98. The equatorial belt. — If we imagine a sphere con- 
structed on the polar diameter of the earth, the difference be- 
tween the sphere and spheroid will be a sort of shell or ring, 
thirteen miles thick at the equator, and growing thinner on 
every side to the poles. This is sometimes called the equa- 
torial ring or belt of the earth, and it produces sensible effects 
on the earth's relations to the moon and sun. 



99. Weight and density of 
the earth. — The earth's mass, 
and therefore its density, can 
be obtained by comparing the 
effects produced upon a plumb- 
line, by the earth and a moun- 
tain of known weight. Let M 
(Fig. 34) be an abrupt moun- 
tain situated alone on a plain, 
and let a station, B, be selected 
on the north side of it, and an- 
other, D, in the same meridian, 
on its south side, for measuring 
the zenith distances of stars. If 



Fig. 34. 




* B- 




60 DIUKtfAL KOTAT10N. 

the mountain were not present, the plumb-line of the zenith 
sector would hang in the lines B and D, and would mark E 
and G as the zeniths of the stations. But the attraction ui 
the mountain draws the plumb-line toward it, so as to joint to 
the false zeniths E' and G'. When the star S, therefore, 
culminates, its apparent zenith distance, SE', is measured at 
one station, and at another culmination, SG' is measured. The 
difference, SE' — SG', is the distance between the apparent 
zeniths. The distance, EG, between the true zeniths, is the 
same as the difference of latitude between the stations B and 
D. Let a trigonometrical survey, therefore, be made around 
the mountain, and thus the arc BD, or its equal EG, be found. 
E'G' — EG = the sum of the two angles by which the plumb- 
line is drawn from a vertical position at the two stations. 
The volume and density of the mountain being measured, and 
the angle being found, as above, by which it draws a plumb- 
line from a true vertical, we have the means of determining 
the mass of the earth. And, as its volume is known, its 
density is inferred. Observations of this kind were made near 
Mount Schehallien, Scotland, by Dr. Maskelyne, who found 
the deviation of the plumb-line to be a little more than 6". 

The mean density of the earth, as deduced from a great 
number of results, obtained by this and other methods, is 5.46, — 
that is, the earth, as a whole, is 5.46 times the weight of the 
same volume of water. Calling the weight of a cubic foot of 
water 62^ lbs., the weight of the earth is somewhat more than 
6,000,000,000,000,000,000,000 tons. 

1 OO. Proofs of the earth? s diurnal rotation. 

1. To suppose the earth to rotate eastward on its axis, is the 
only reasonable way of explaining the fact, that all the mill- 
ions of fixed stars, at various and immense distances from us, 
in large and in small circles of the sphere, perform their ap- 
parent revolutions about us in precisely the same length of time 
— viz., one sidereal day. 

2. "Without supposing the earth to rotate on its axis, we can 
not account for the oblate form, of the waters of the ocean. 
Whatever form the solid parts might have, the movable portion 
would be spherical, if the earth were at rest. Moreover, the 



ROTATION OF THE EARTH. 61 

degree of oblateness is exactly that which is required on a 
sphere having the diameter and mass of the earth, if it be sup- 
posed to rotate once in 24 hours. 

3. The weight of a body at the equator, compared with that 
at the poles, is too small to be wholly accounted for by in- 
creased distance. Centrifugal force, arising from rotation, can 
alone explain the remaining difference. 

4. A body dropped from a great height strikes further east 
than the vertical line in which it began to fall. If the earth 
rotates, the top of a tower moves faster than the base ; and 
therefore a body let fall from the top, retaining the east- 
ward motion of that point, will strike further east than the- 
base. At the equator, this distance would be near 2 inches, 
for a fall of 500 feet. Numerous experiments on the fall of 
bodies through great distances have been very carefully made 
by different individuals, and in different latitudes. And they 
all concur in proving that a body in falling deviates from a 
vertical line toward the east. 

5. It is jxroved by the vibrations of a pendulum that the 
earth rotates eastward. Let us suppose a weight to be sus- 
pended by a long fine wire, and then made to vibrate in a 
plane. The plane in which the wire and weight move is ver- 
tical, and passes through the point of suspension. The weight 
itself may be considered as describing a straight horizontal line. 
On account of inertia, the weight tends to keep always in the 
same line, or (if the point of suspension be moved) in a line 
parallel to itself. And it will always remain strictly parallel tc 
itself, provided it can at the same time remain horizontal, and 
in a vertical plane passing through the point of suspension. 

Thus, if at the equator the weight be made to vibrate north 
and south — that is, in the plane of a meridian— it will continue 
to do so without deviation, as the earth rotates eastward, be- 
cause it will thus remain moving horizontally in a plane which 
passes through the point of suspension, though that plane is 
continually changing. In this case, the lines in which the 
weight vibrates are all parallel among themselves. 

If the experiment be tried at the pole, and the weight be 
made to vibrate in the plane of a certain meridian, the point of 
suspension does not move from its place, but only revolves in 



-62 FORM OF THE SUN. 

it ; and while the earth revolves 15° per hour, the weight pre- 
serving its own plane of vibration, will seem to shift that plane 
15° per honr in the contrary direction, keeping pace with the 
stars in their diurnal motion. 

At localities between the equator and the pole, the line oi 
vibration remaining horizontal, and in a vertical plane which 
passes through the point of suspension, can not at the same time 
preserve its parallelism. But it will come as near fulfilling 
this condition as possible. Its north extremity will deviate 
eastward from the meridian more or less, according as it is 
nearer the pole or the equator. It is proved that the deviation 
per hour is to 15° as the sine of latitude to radius. 

When experiments are performed with sufficient care, it is 
found that the pendulum actually deviates eastward from the 
meridian, and at a rate corresponding well with the calculated 
result. The pendulum thus furnishes evidence that the earth 
rotates on its axis. 

The above is known as Foucault's experiment. 

6. It will be seen hereafter that the motion of the equinoc 
tial points toward the west, called the precession of the equi 
noxes, affords an independent proof of the earth's diurnal 
motion. 



CHAPTEK VII. 



THE SUN. — SOLAR SPOTS. — CONDITION OF THE SUN'S 
SURFACE. — THE ZODIACAL LIGHT. 

101. The form of the sun. — The disk of the sun is always 
circular. And, as it presents all sides toward us in its rotation, 
we infer that its form must be spherical. But since it rotates 
on an axis, and its surface is hi a fluid state, it might be ex- 
pected to reveal a spheroidal form. The reasons why it does 
not are, that the force of gravity on the sun is very great, and, 
in consequence of the slowness of its rotation, the centrifugal 
force is small. It appears by calculation that the angle sub 
tended by the equatorial and the polar diameters can not differ 



DIMENSIONS OF THE SUN. 63 

from each other, except by a small fraction of a second. Its 
oblateness is, therefore, too slight to be perceived. 

102* Distance of the sun, and size of the earth's orbit.—* 
The sun's horizontal parallax is 8."848. Therefore, the distance 
of the sun from the earth is found (Fig. 4) by the proportion. 

sin 8."848 : rad : : 3962.802 : 92,381,000-; 

which is the distance in miles from the earth to the sun. 

The circumference of the earth's orbit, or the distance trav 
eled by the earth each year, is 

92,381,000 x 2?r = 580,447,000 miles. 

1 03. Velocity of the earth on its axis and in its orbit coin- 
pared. — In the diurnal motion, a place on the equator describes 
nearly 25,000 miles in 24 hours — that is, more than 1,000 
miles per hour, or about 17 miles in a minute. In the annual 
motion, the earth describes 580,447,000 miles in 365J days, 
thus passing over a distance of 1,589,000 miles each day; 
which is about 1,103 miles in a minute, or 18.393 miles in a 
second. The earth's velocity in its orbit is about 65 times as 
great as that of the equator in the diurnal motion. 

1 04. To find the dimensions of the sun. — The angle sub- 
tended by the sun's diameter may be measured by instruments. 
Let AES (Fig. 35) equal one-half the measured angle. Then 
we have rad : sin AES : : ES : AS, the semi-diameter of the 
sun. As the sun's mean apparent semi-diameter is 16' 2", and 
ES is 92,381,000 miles, we find the sun's radius near 430,855, 
and therefore its diameter 861,710 miles. 

Fig. 35. 




The sun's diameter is about 109 times that of the earth. 
And, since spheres vary as the cubes of their diameters, tho 
volume of the sun to that of the earth is as 

109 3 : l 3 : : 1,295,000 : 1, nearly. 



64 DIURNAL ROTATION OF THE SUN. 

1 05. The surfs mass and density. — It is found, by methodi 
to be described hereafter, that the sun does not exceed the 
earth in mass nearly so much as it does in volume. While 
the volumes are as 1,295,000 : 1, the masses are about as 
326,800 : 1. 

The density of the sun, therefore, is to that of the earth a? 
326,800 : 1,295,000 : : 1 : 4, nearly. 

106. Force of gravity at the surface of the sun. —When 
the relative masses and diameters of bodies are known, it is 
easy to find the relative force of gravity on their surfaces. For 

G oo -^- (TS"at. Phil., Art. 16), where G represents gravity, Q 

the mass of the body, and D its semi-diameter. Let W repre- 
sent weight at the earth, and W at the snn, and we have 

W : W : : -1 : S *^° : : 1 : 27.5. Hence, the weight of a body 

at the sun is 27.5 times as great as at the earth, and a body 
would fall 442 feet in the first second of its descent. 

107. Diurnal rotation of the sun. — By observations on the 
solar spots, it is found that the snn rotates on its axis nearly in 
the same direction in which the earth revolves about the sun. 
In general, a spot which appears on the edge of the disk passes 
across, then disappears, and afterward reappears in the same 
place as at first in 27^ days. If the earth were at rest, this 
wonld be the period of the sun's ^ 36 
rotation on its axis. But, as the 
earth revolves in nearly the same 
direction in its orbit, the appa- 
rent rotation of the snn is longer 
than its real rotation. In Fig. 
36, suppose the earth to be sta- 
tionary at E, and that a spot on 
the sun appears on the diek at 

A. Then, after passing through 

B, D, H, it will appear again at 
A, at the end of one revolution. "^e — "^ F 
But, if the earth in the mean time moves on to F, the*> the 




APPEARANCE OF THE SOLAR SPOTS. 65 

spot must pass over AB, in addition to one revolution, before 
it will be seen on the edge of the disk. As EC is perpendicu- 
lar to AD, and FC to BH, the corresponding arcs on the two 
circles are obviously similar. Therefore, EGE + EF : EGE : i 
ADA + AB : ADA. Instead of the arcs, we may use the 
times of describing them ; and then we have 1 year -f 27^ 
days : 1 year : : 2T| days : 25 days, 8-J hours, which is the 
period of the sun's rotation. Appendix A. 

108. Position of the sun's equator. — If the solar spots al 
ways described their paths across the disk in apparent straight 
lines, it would be inferred that the sun's equator coincides 
with the plane of the ecliptic. But these lines appear straight 
only twice in the year, near the middle of June and of December. 
At other times, they appear as semi-ellipses, having the greatest 
breadth in March and September. The earth, therefore, passes 
the plane of the sun's equator in June and December. The 
inclination of the sun's equator to the plane of the ecliptic is 
found to be about 7i°. 

109. — Appearance of the solar spots. — On examining the 
sun's disk with a telescope, there is usually seen a greater or a 
less number of dark spots, differing from each other in form 
and size, and each spot generally consisting of two distinct 
parts, called the macula, or nucleus, and the umbra. The 
macula is black, of irregular form, and commonly surrounded 
by the umbra, which has a lighter shade. The two parts of 
the spot do not often shade into each other, but are each 
marked by a sharp, though irregular outline. If watched from 
day to day, they are seen not only to move slowly across the 
disk, as already stated, but they change their form and general 
appearance. A large spot sometimes divides into two or more 
smaller ones ; and again a group unites into a single large spot. 
Sometimes a spot diminishes and disappears, first the macula, 
then the umbra. The reverse also happens — a spot is seen in 
the midst of the disk, where there was none the day before. 
Though only a few are commonly in sight at once, yet they have 
been, in some instances, counted by tens and even hundreds. 
Very rarely a spot is so large as to be seen by the naked eye. 

5 



bb RELATION OF SPOTS TO SURFACE LEVEL. 

Figure 37 (lower part) shows two views of the same grou|; 
as seen July 9th and 11th, 1 844. 

Fig. 37. 




110. The spots are at the surface, and limited to a northern 
and a southern zone. — Each spot appears on the disk during one- 
half the time of its entire revolution. It must, therefore, be at 
the surface, and not at any distance from it. For, if it revolved 
at any distance from the surface, as in the orbit abc (Fig. 35),, 
then it would be seen on the disk only from a to h, which is 
less than half its orbit. 

But the spots do not pass across all portions of the disk; 
their paths are limited to a zone which extends not more than 
35° on each side of the equator ; and with very few exceptions,, 
they lie in the outer, rather than the central parts of this zone. 
Spots are very rarely seen within the zone lying between 10° 
of north and south latitude; and still more rarely in the polar 
zones above latitudes 35° north and 35° south. The macular 
zones, as they are sometimes called, are represented in Figure 
37, limited by the dotted lines, EQ being the equator. 

111. Relation of the spots to the surface level. — If the spots 
were flat surfaces on the same level with the general surface 
of the sun, then all their parts would be foreshortened alike, 



THE RECEIVED THEORY. 67 

when near the edges of the disk. If they were elevated ob- 
jects, as mountains, rising above the solar atmosphere, then the 
umbra nearest the edge of the disk would be hidden by the 
darker part, and on the edge the spot would appear as a pro- 
tuberance. 

But it is proved, by multiplied observations, that the spots 
must be degressions below the general surface, and the macula 
a deeper depression than the umbra. For, as a spot approaches 
the edge, while it is foreshortened by perspective, the umbra 
furthest from the edge disappears first, and then the macula 
itself, while that part of the umbra nearest the edge is still in 
sight. As a spot comes from the edge toward the central part 
of the disk, the order of appearances is* reversed. These 
changes are indicated in Fig. 37, upper zone. Appendix B. 

IIS. The general surface, — The luminous part of the sun's 
surface is not uniform, nor at rest. Every portion of it is mi- 
nutely mottled by spots and streaks of unequal illumination. 
These are called facidce. And continued observation shows 
that these faint inequalities are also undergoing incessant 
changes. The faculse are most strongly marked, and indicate 
the greatest agitation of surface, where a spot is about to ap- 
pear, or where one has recently disappeared. Appendix C. 

113. The received theory. — No theory so well explains the 
telescopic appearances of the sun, as that which in substance 
was proposed by Sir William Herschel, in 1801. Whatever 
may be the condition of the central mass, the external surface, 
called the photosphere, consists of gas in an incandescent state, 
while below it, within the solar atmosphere, is a cloudy 
stratum, less luminous than the outer surface. Whenever, 
from any cause, a rent is made in the photosphere, the less 
luminous stratum below is seen through it, as the umbra of a 
spot; and a smaller rent in the lower stratum reveals the 
denser and darker part of the sun, as the macula of the same 
spot. The strata in which the rents occur are in a gaseous 
condition ; for the constant motions going on in the outlines 
of the spots, forbid the supposition that they consist of solid 
matter ; and the extreme rapidity of these motions, often more 



68 THE ZODIACAL LIGHT. 

than 1,000 miles per day, is inconsistent witA the idea thai 
they are liquid. 

114. The tody of the sun not necessarily dark. — The very 
dark appearance of the macula may be due to its strong con 
trast with the intense illumination of the general surface. 
For it is found by experiment that the brightest artificial light 
which has been produced, if placed between the eye and the 
sun, appears as a dark spot compared with the solar surface. 

115. Cause of the spots. — Sir John Herschel has suggested 
that there are reasons for considering the equatorial regions of 
the sun to be more heated than the other portions, so that there 
are currents in the solar atmosphere analogous to the trade- 
winds on the earth. Resulting from these currents, he sup- 
poses that occasional local winds are produced, rotating on a 
vertical axis, and rending the atmosphere and clouds by their 
centrifugal force. The ruptures thus occasioned are the spots 
on the sun. 

This supposition derives considerable plausibility from the 
considerations, that the spots are limited to narrow zones a little 
distance from the equator; that they sometimes differ from 
each other in their motions across the disk ; and that, in a few 
instances, they have shown signs of rotation about their own 
centers. Appendix D. 

116. Periodicity of the spots. — The number and size of 
spots vary exceedingly in different years. Sometimes for days 
and weeks none are to be seen ; and again, for many months, 
the disk is never free from them. It is noticed, of late years, 
that their frequency alternately increases and decreases during 
a period of 10 or 11 years. The years in which the greatest 
number has been seen of late, were 1870, 1882. And those in 
which there were fewest, were 1867, 1878. Appendix E. 

117. The zodiacal light. — This name is given to a faint, ill- 
defined light, extending along the zodiac, either in the west, 
after sunset, or in the east, before sunrise. It so much resem- 
bles the twilight, that it is not ordinarily noticed, because it 



KEPLER'S LAWS. 



69 



appears as a mere upward extension of it. It is projected on 
the sky as a triangle, inclined to the horizon at the same angle 
as the ecliptic (Fig. 38). In the 
evening it is best seen at the 
season when the ecliptic is most 
nearly perpendicular to the ho- 
rizon, after twilight has ceased. 
It is therefore most conspicu- 
ous at evening in the month 
of February. When the air 
is clear, and there is no moon, 
it is visible till after 9 o'clock. 
For a like reason, the best 
time for seeing it before morn- 
ing twilight is the month of 
October. The apparent extent 

breadth and 

increased by 



of it, both in 
height, is much 
indirect vision. 




118. Its nature. — There has been much speculation rela- 
tive to the nature of the zodiacal light. But astronomers gen- 
erally regard it as a nebulosity attending the sun, and extend- 
ing beyond the orbits of Mercury and Venus, and even beyond 
the orbit of the earth. 



CHAPTER VIII. 



KEPLER S LAWS. — THE LAW OF GRAVITATION. 



119. Statement of Kepler's laws. — From a long and labo- 
rious examination of the recorded observations of Tycho Brahe, 
Kepler deduced three laws relating to the movements of the 
planets about the sun. They are hence called Kepler's laws, 
and may be stated as follows. 

1. The areas described about the sun by the radius vector of 
an orbit, vary as the times of describing them. 



70 



LAW OF AREAS. 



2. The orbit of every planet is cm ellipse, having the sup. in' 
one focus. 

3. The squares of the periodic times of the several planets 
vary as the cubes of their mean distances. 

To render the language of the third law strictly correct, the 
cube of the distance should be divided by the sum of the masses 
of the sun and planet. But the mass of even the largest planet 
is so small, compared with the sun, that the omission intro- 
duces an error which is scarcely appreciable. 

Kepler established these three laws as facts in the solar 
system ; but Newton afterward demonstrated, by mathematical 
reasoning, that they are necessarily involved in the laws of in- 
ertia and gravitation. 

120. Areas described by the radius vector. — Whatever path 
a body describes under the influence of a projectile and a cen- 
tripetal force, the areas described about the center of force vary, 
as the times of describing them. 




Let S (Fig. 39) be the center of attraction, and suppose the 
projectile force in the line YE, to be such as to cause the body 
to pass over the equal spaces PQ, QR, etc., each in a certain 



LAW OF VELOCITY IK AN ORBIT. 71 

emit of time. When the body reaches Q, let the action 
toward S be sufficient to move it over QY in the same time 
in which by the original impulse it would describe QR. Then 
it will in the same time describe the diagonal QC of the par- 
allelogram. Jo£n ES and CS. The triangles QSC and 
QSR are equal ; but QSR = QSP ; .\ QSC = QSP ,— that is, 
the areas described in the first and second units of time are 
equal. In like manner, by supposing a second action toward 
S to occur at C, a third at D, etc., it is proved that QCS, CDS, 
DES, etc., which are described in equal times, are equal. This 
is true, however small the unit of time between the successive 
actions toward S, and is therefore true when the central force 
acts incessantly and causes curvilinear motion. As all the 
areas are equal, which are described in the several units of 
time, therefore the areas vary as the times. 

As the diagonal of each parallelogram is in the same plane 
with its two sides, it is obvious that the whole orbit lies in one 
and the same plane. 

Conversely, if areas described about a point vary as the 
times, the deflecting force acts toward that point. For 
PSQ = QSE, as before (Fig. 39) ; and by supposition, PSQ = 
QSC ; .% QSC - QSR ; hence CR is parallel to QS, and QC is 
the diagonal of a parallelogram, whose side QY, in which the 
deflecting force acts, is directed toward S. 

Since it is an established fact, agreeably to Kepler's first law, 
that the radius vector of each planetary orbit describes areas 
about the sun, which vary as the times; therefore, the cen- 
tripetal force, acting on the planets, is directed toward the sun. 

121. The law of velocity in an orbit. — The velocity at any 
point varies inversely as the perpendicular from the center of 
force to the tangent at that point. 

Let ST (Fig. 39) be perpendicular to PQ ; then the area 

rSPQ =iPQ x SY, which varies as PQ x SY; ,\ PQ oo ^$. 

But PQ oo Y, the velocity at P ; and the area SPQ is constant; 

•"*- "V* °° oyj or the velocity varies inversely as the perpendicu- 



72 



LAW OF GKAVITATICXN. 



lar from S, upon the line in which the body is moving; in 
other words, upon the tangent of its path, if it describes a 



curve. 



122. Law of gravitation in an orbit, as related to dis- 
tance. — If a body describes an elliptical orbit, by a centripetal, 
force which acts toward the focus, that force varies inversely a& 
the square of the distance. 



Fig. 40. 




Let the body be at M (Tig. 40), and MF the radius vector at 
that point. Let MO be the radius of curvature at M, and. 
therefore perpendicular to the tangent ; and suppose M^N" to be 
an infinitely small arc described in a given small portion o£ 
time. Draw FP perpendicular to the tangent MP, KK to 
FM, and NH to MO ; then PFM, MHI, KNI are similar tri 
angles. ME", considered as a straight line, is described by the 
joint action of the centripetal force in the line MI, and the 
projectile force which is parallel to IN. The motion in MI 
may be regarded as uniformly accelerated, because in the in- 
finitely small time of describing it, the centripetal force may 
be considered constant. Hence, 2MI may be taken as the 
measure of the centripetal force f (ISTat. Phil., Art. 28). 

Therefore, f c© MI. It is to be proved that MI qo r^. 



LAW OF GKAVITATION. 73 

123. By similar triangles, MI : MH : : XI : NK; 

NI 

flow, the chord MN is a mean proportional between tha 

TVTNT 2 
versed sine MH and the diameter 2MO ; or MH = oiuTj » 

Nil 2 
but, as the arc is infinitely small, NH = MIST ; .\ MH = o]v/r(V 

Again, the versed sine MH, and therefore HI, is infinitely 
small compared with NH, and NI may be substituted for NH ; 

••• MH = Sro- 

124. Now it is shown in conic sections, that r 

mKJ 2 V FP/ ' 

FM NI 

therefore, since by similar triangles -^p = -Sxt?? 

M0 =*ffiY 

2VJSTK/ 
Substituting this for MO in the equation for MH above, we have 

MH = m . 

^> . NI 

Hence, in the equation for MI we have 

EK 3 NI 1_ T _■ 
MI = — ^ T x ^^ = - NK. 

j? . ]NI M jp 

Now, the sector FMN is measured by JFM . NK ; . • . NK = 

* Jackson's Conic Sections. The same may be derived from Coffin's Conic 

(FM MV) 3 
Sections, Pr. V., Curvature, R 2 or MO = v ■— — - , a and b being tlie semi- 

1 3 3 

axes; .'. MO = — x (FM. MV) 2 . Multiply by (& 2 ) 2 , and divide by its equal 
(FP VL)s ; JkenMO = L (^^) 2 = °- (|^j 2 , since FMP and VML 
are similar. But - = |; ,. MO = f (— )* = f ( p --) . 



74 LAW OF GRAVITATION. 

2FMN , AT1Z2 4FMN 2 __. 4FMN 2 _ t ■_ 
-^j- ; and NK 2 = -j^-; .-. MI = — w , But as the 

areas described by the radius vector vary as the times, FMN ia 
constant. Therefore, 

MI(=/)co^; 

that is, the centripetal force in the orbit varies inversely as the 
square of the distance. 

125. Applicable to every conic section. — It is thus proved 
that, in any elliptical orbit described about the focus as the 
center of attraction, the intensity of that attraction varies in- 
versely as the square of the radius vector. As there is nothing 
in the foregoing demonstration to limit the conclusion to the 
orbits which are nearly circular, like those of the planets, we 
are at liberty to apply it to orbits of extreme eccentricity, as 
those of the comets. And it is proved by Newton, in his Prin- 
cipia, that the same law of force is necessary, in order that a 
body may describe any one of the conic sections about its focus 
as the center of attraction. 

126. Law of gravitation as to distance, in different or- 
bits. — And not only does this law prevail in all parts of any 
one orbit, but it is true also that all the different bodies of a 
system, describing orbits about the same center of force, are 
urged toward that center by attractions which vary, from one 
orbit to another, inversely as the square of the distance. 

Let a be the semi-major, and b the semi-minor axis of any 
elliptic orbit. Then a is the mean distance of all points of the 
orbit from the focus. By a rule of mensuration, the area of 
the ellipse = nab. If s = the area described by the radius 
vector in a unit of time, as one second, and t = the number of 
seconds in the whole period of revolution, then the ellipse also 

= ts. Therefore, nab = is ; and t = — ; and f = — — . By 

27 2 7 2 

Kepler's third law (Art. 119), tfao a 3 ; .*. —5- 00 a z ; ,\ — op s\ 

s a 

But, because the semi-parameter ^- is a third proportional to 



LAW OF GRAVITATION. 



75 



the semi-axes a and £, — = ^ ; 



foes. 



Hence, substituting 



% for s 2 ,— that is, FMN 2 ,— in the equation for MI (Art. 124), we 



find MI 



4FMN 2 = 2/y 2 

^.FM^FM 2 ' 



.-./.oo 



Or, the 



jp . FM 2 jp. FM 2 FM 2 ' " J " FM 2 * 
force varies inversely as the square of the distance, in different 
orbits, as well as in different parts of the same orbit. 

The satellites which revolve about the planets are found to 
conform to Kepler's laws, and therefore the force which urges 
them toward their respective primaries varies in each case in- 
versely as the square of the distance. 

127. Law of gravitation within small distances. — But the 
inquiry still remains, does the law of gravity, as demonstrated 
in the foregoing articles, hold good at the smallest distances 
also ? For example, do the tendencies of bodies resting on the 
earth, and of those elevated in the air, and of the moon toward 
the earth's center, come under the same general law ? This is 
the very question which presented itself to the mind of New- 
ton, after he had discovered that the force which deflects the 
planets from their lines of motion toward the sun, varies in- 
versely as the square of their distance from it. As he noticed 
the fall of an apple, the inquiry arose, may not this fall be of 
the same nature as the lending of the moon's path toward the 
earth, and may not the force in the two cases be as the squares 
of the distances inversely % 

The distance through which the 
moon actually descends in one second 
may be represented by Ka (Fig. 41), 
A o being the arc described in the same 
time. For, as the moon was going 
toward B, it would not have deviated 
from the line AB, if some force had 
not turned it aside. This influence 
must be directed toward the earth, E, 
because it is about E that the radius 
is known to describe areas propor- 
tional to the times (Art. 120). There*- 




76 LAW OF GKAVITATTON. 

fore, Bh, or the versed sine Aa (which may be considered equal 
to it), is the distance fallen through in one second. Now, the 
circumference of the moon's orbit, divided by the number of 
seconds occupied in describing it, gives the arc Ah. This arc 
and its chord may be considered the same, and by geometry 
we have 2 AE : Ah : : Ah : Aa = 0.0535 of an inch. 

At the surface of the earth, a body falls 1 6^ feet in the 
first second. On the supposition that gravity varies inversely 
as the square of the distance, we find the fall in one second at 
the moon, by the proportion, the square of the moon's dis- 
tance : square of the earth's radius :: 16^ feet : 0.0536 of an 
inch, agreeing very accurately with the distance which the 
moon actually falls from a tangent in one second. Therefore, 
a body falling at the surface of a planet, and a satellite revolv- 
ing about it, are both subject to the same law of centripetal 
force. 

128. The law prevails throughout the solar system. — As 
will appear hereafter, there are numerous disturbances pro- 
duced upon the motion of each body in the system by the 
attraction of every other. Every one of these disturbing influ- 
ences is measured, by applying the law of distance already men- 
tioned. If a planet or comet moves toward a plauet for a 
certain length of time, it is accelerated ; and its acceleration is 
greater, as the square of the distance is less ; and it is retarded* 
according to the same law, when departing from it. 

129. The law of gravitation, as related to the quantity of 
matter. — The force of gravity varies directly as the quantity ot 
matter. In Mechanics, we infer the existence of this law from 
the fact that all bodies, light and heavy, and of every kind ot 
material, fall with equal velocity toward the earth. So, in the 
solar system, a planet and all its satellites, when at equal dis- 
tances from the sun, are urged toward it by forces proportional to 
their masses, or they could not maintain their mutual relations 
as they do. And it is found that every disturbing influence in 
the system is accounted for only by applying both parts of the 
law of gravity — that it varies directly as the quantity of matter % 
and, inversely as the square of the distance. 



PATHS OF PROJECTILES. 



77 



130. Paths of projectiles considered as orbits. — When a 
6tone is thrown, or a ball is fired, its path (undisturbed by the 
atmosphere) is part of an elliptic orbit, one of whose foci is at 
the center of the earth. In Mechanics, the path of a projectile 
is proved to be a parabola (Nat. Phil., Art. 44) ; but, in that 
demonstration, the vertical lines were assumed to be parallel 
to each other, and the force of gravity a constant force, neither 
of which is strictly true. Knowing the distance and period of 
the moon, the time in which a projectile would complete its 
revolution if. found by Kepler's third law. Any force, which 
man could apply, would carry the lower extremity of the orbit 
so little beyond the center of the earth, that the mean distance 
might be. called one-half the radius of the earth. Therefore, 
calling the moon's distance 60 radii, and its period 27J days, 
we have (60) 3 : (£) 3 :: (27J) 2 : x% from which x is found to be 
about 31 minutes. Every projectile, then, if it were free to 
complete its orbit unobstructed, and according to the law of 
gravity which prevails outside of the earth, would make an en- 
tire revolution, and return to its place, in about half an hour. 



131. Effect of increased velocity of projection. — Suppose 
that P (Fig. 42) is a point near the earth, ADE, and that the 
velocity of projection, in the direction PB, is so greatly in- 
creased that the projectile strikes the earth at D. By a still 
greater increase of velocity 
it might meet the earth at E. 
In these cases the earth's 
center would be in the most 
remote focus of the orbit. 
But if we suppose the velo- 
city so much increased that 
the centrifugal force just 
equals the force of gravity, 
then the body would de- 
scribe the circular orbit PFG 
(Art. 90). As the mean dis- 
tance now equals the radius 
of the earth, the time of revolution is found, by Kepler's third 
law, to be lh. :4m. 39s. Any increase of the velocity of pro- 




78 MOTIONS OF SUN AND PLANET. 

jection beyond this will again produce an ellipse, as PK. 
whose nearer focus is at the earth's center. And we can 
imagine the velocity increased till the ellipse becomes one ol 
extreme eccentricity, and then changes into the branch of a 
parabola, and then of a hyperbola, in which last cases the body 
will never commence a return toward the earth. 

132. Orbit motion and diurnal rotation by one impulse.— 
If we suppose the projectile motion of the earth, or any other 
planet, to have been produced by a single impulse, that im- 
pulse may also have caused the diurnal rotation of the body. 
If the impulse had been directed in a line passing through the 
center of gravity of the planet, then it would have caused a 
progressive motion without rotation on an axis. But, if the 
line of impulse did not pass through the center of gravity, 
there would be rotation as well as progression. It has been 
calculated that the two existing rotations of the earth might 
have been produced by one impulse, applied in a line which 
passes 24 miles from the earth's center, on the side most 
remote from the sun. 

Had it been directed through a point lying on the side 
nearest the sun, the diurnal motion would obviously have been 
retrograde. 

133. Motions of sun and planet, resulting from an impulse 
given to the planet. — Suppose that the sun at S (Fig. 43), and 
the earth at E, mutually attract each other, and that an im- 
pulse is given to E in a line perpendicular to ES. S can not 
remain stationary and E revolve about it ; for it is proved (Nat. 
Phil., Art. 89) that their center of gravity will move precisely 
as the sum of the bodies would move if united at the center, 
and the same impulse were applied to them. Suppose, for the 
sake of simplicity, that the weights of the bodies and the 
strength of the impulse are so related that the center, C, will 
pass over each unit of space, Ga, ab, bo, etc., while E advances 
45° in a circle about the moving center. Then, when the 
center is at a, E is at 1, 45° from a perpendicular at a. But S 
must be on the opposite side of a, and as far from it as from 
before. Therefore, by the impulse given to E, and the mutual 



A PLANET AT APHELION OR PERIHELION. 79 

Attraction between E and S, the latter has been drawn along 
from S to 1/. Again, when the center is at b, E is at 2, and S 
at 2'. While E was on the upper side of CA, S was drawn 
toward that line, and now crosses it, and by its inertia con- 
tinues upward, although E is now below the line. In this 
manner the bodies revolve about the moving center, describing 
circles relatively to that, but curves of a totally different char- 
acter in space. These curves are always some variety or other 
of the class of curves called epicycloids. In the case repre- 
sented in the figure, the planet describes an epicycloid which 
forms a series of loops, intersecting its own path at every revo- 
lution, while the path of the heavier body is of a waving form. 
The body E retrogrades on the lower part of the loop from 3 
to 5, while S advances continually, but with unequal velocities, 
each body being alternately drawn forward and held back by 

the other. 

Fig. 43. 

-E 




The only way in which two separate bodies could be made to 
rotate about a fixed center of gravity, would be to give an 
equal impulse to each body, and in opposite directions. Two 
such forces would constitute a couple (Nat. Phil., Art. 54), 
whose effect is to produce rotation merely. 

134. Why a planet at aphelion begins to return, or at peri- 
helion begins to depart. — It might be thought that a planet at 



80 



PRECESSION OF EQUINOXES. 



its aphelion, C (Fig. 44), being less attracted toward the sun 
than at any other point, wonld continue to withdraw, instead 
of commencing to return ; and that when at its perihelion, G-, 
being more attracted than else- 
where, it would continue to ap- 
proach till it falls to the sun. The 
reason why a planet begins to re- 
turn after reaching the aphelion is 
to be found in its diminished ve- 
locity. As the plauet recedes 
through. H, K, and A, the centrip- 
etal force toward S draws it back, 
and causes continual retardation, 
till at C the velocity is so much 
diminished that the attraction of S, 
though less than elsewhere, is still 
sufficient to curve the path so that it falls within a circle about 
the centre S, and the planet begins to approach the sun. 

Again, as the planet passes through D, E, and F, the at- 
traction toward S partly conspires with its inertia, and it is 
continually accelerated, till, at G, its velocity has become so 
great that its path strikes outside of a circle about the center, 
S, and it begins again to depart as before. 




CHAPTEK IX. 



PRECESSION OF EQUINOXES. — NUTATION. — ABERRATION OF 
LIGHT. — APSIDES OF THE EARTH'S ORBIT. 



135. Precession of equinoxes described. — The points in 
which the equator intersects the ecliptic on the celestial sphere 
are not stationary, but have a slow retrograde movement — that 
is, they revolve from east to west. The sun, therefore, in its 
annual progress eastward, crosses the equator each year a little 
further west than it did the year previous This motion is 



CAUSE OF PEECESSION. 8 J 

ealled the precession of the equinoxes, either because the time 
of the equinoxes precedes the time in which the sun would have 
passed them if they had remained at rest, or because, in 
the daily transit of the meridian, the equinoxes precede those 
stars which crossed at the same time with them the preceding 
year. 

The equinoctial points retrograde about 50i A ' each year. At 
this rate, it will require 25,800 years to make a complete circuit 
of the heavens. 

136. Signs of the ecliptic displaced from the signs of the 
zodiac. — The want of coincidence between the signs of the 
ecliptic and the signs of the zodiac was noticed (Ait. 61). They 
coincided at the time the division was made, about 2,000 years 
ago ; and the precession daring this period has moved the equi- 
noxes backward 2,000 x 50i" = 28°, nearly. Hence, Aries of 
the zodiac almost coincides with Taurus of the ecliptic, Taurus 
of the zodiac with Gemini of the ecliptic, etc. 

137. Motion of the north and, south poles. — Considering 
the plane of the ecliptic as fixed, its poles of course occupy 
fixed positions among the stars. But this is not true of the 
poles of the equator. Their distance from the polefi of the 
ecliptic is equal to the obliquity of the two circles — that is, 
23° 27 / . As this angle remains nearly constant, and the points 
of intersection move around westward, the poles of the equator 
must likewise move round those of the ecliptic in the ^ame 
direction, and occupy the same period, 25,800 years in com- 
pleting their revolution. The north pole of the equator is row 
near the star in Ursa Minor, known as the pole-star. Accord- 
ing to the earliest catalogues, the pole was 12° distant from the 
pole-star. It is now somewhat more than 1° distant, and will, 
at the nearest, pass within \° of it. In about 13,000 years the 
pole will be on the opposite side of the pole of the ecliptic, near 
the bright star a Lyrse, which will then be the pole-star. 

138. Cause of precession. — The precession of the equinoxes 
is a disturbance produced by the sun's and moon's attraction 
upon the equatorial ring of the earth, as it rotates on its axis. 

6 



82 



CAUSE OF PKECESSION. 



The sun being in the ecliptic, while the equatorial ring is inclined 
23° 27' to it, the sun's attraction is oblique to the plane of the 
ring; and one component of this force is perpendicular to the 
ecliptic. In most positions of the ring in relation to the sun, 
this component acts on one part to press it towards the ecliptic, 
and on another part to move it from the ecliptic. But the first is 
in excess ; so that, on the whole, the ring tends to turn on the 
line of equinoxes towards the plane of the ecliptic. And this 
tendency, compounded with the inertia of the ring in its diurnal 
rotation, moves the equinoxes backward. 

Fig. 45. 




Let EC (Fig. 45) represent the plane of the ecliptic, ana 
QR the equatorial ring of matter. A particle, A, of the ring, 
by its inertia of rotation, tends to move toward T in the plane 
QR. Let AB represent this force, and AF the pressure toward 
EC, produced by the sun ; then the resultant will be the diag- 
onal AD, shifting the equinox back to T'. All the particles 
are subjected to this influence, except at the moment (each day) 
of crossing T and — , so long as the sun itself is not in the line 
T=^ produced, which occurs in March and September. The 
effect is then interrupted for a time. 

As the moon is always near the ecliptic — sometimes on 
one side of it, and sometimes on the other — its action on the 
whole conspires with that of the sun. And as it is compar- 
atively near, though it is so small a body, its effect is more 
than twice as great as that of the sun. The planets produce a 



THE TROPICAL AND SIDEREAL YEAR. 83 

very minute effect on the ring, tending to diminish the amount 
of precession. The joint effect of all the bodies mentioned is, 
as stated above, 50J". 

139. Law of composition of rotations. — The case of pre- 
cession of equinoxes is classed under the general law for the 
composition of two rotations, which is analogous to that for 
the composition of two rectilinear motions (Nat. Phil., Art. 
38). It may be stated thus : if two forces are applied to a 
body, which, separately, would cause rotation on two different 
axes, their joint action will produce rotation on a third axis 
lying in the plane of the other two, and making angles with 
them, whose sines are inversely as the forces. In precession, 
the earth rotates on the diurnal axis by one force, and the sun 
and moon tend to rotate it on the line of the equinoxes. As 
the latter force is minute compared with the other, the new 
axis is shifted by a very small angle each year from the diurnal 
axis toward the line of equinoxes. And this line slides along 
the ecliptic, so that the two axes remain perpetually at right 
angles with each other. 

The rotascope, a modification of Foucault's gyroscope, may 
be used to exhibit a very perfect illustration of the precession 
of equinoxes. 

140. Cause of the slowness of precession. — If the equatorial 
ring were a separate body rotating about the earth in its own 
plane, its points of intersection with the ecliptic would retro- 
grade very rapidly by the action of the sun and moon. The 
reason why the precession is exceedingly slow is, that while 
the disturbing action is exerted only on the ring, the force 
around the diurnal axis consists of the inertia of the entire 
earth. The ring can not move by itself, but must carry the 
whole mass of the earth with it. 

141. The tropical and sidereal year. — The fact of preces- 
sion shows that the year has two different values, according as 
we reckon from a star or from an equinox. Hence, the side- 
real year is defined to be the period occupied by the sun in 



£4 NUTATION. 

passing eastward around the heavens from a star to the same 
star again ; and the tropical year, the time of passing around 
from an equinox to the same equinox again (Art. 86). As the 
equinox moves westward, the sun reaches it sooner than if it 
were stationary, and thus makes the tropical year shorter than 
the sidereal, by the time required to pass over 50y, which is 
20m. 22.9s. As the tropical year is 365d. 5h. 48m. 46.15s. 
(Art. 86), the sidereal year, therefore, is 365d. 6h. 9m. 9s. 

Though the sidereal year is the true period of the earth's 
revolution about the sun, yet the tropical year possesses by far 
the greatest interest, because it is the period in which the 
seasons are completed. 

142. Nutation. — By precession alone, the pole of the 
equator would move in the circumference of a circle about the 
pole of the ecliptic. But this motion is modified by a minute 
vibration from side to side, as it 
advances, so that the line described Fi g- 46. 

by the pole is a delicate wave lying 
along on the circumference, as rep- 
resented in Fig. 46, where P repre- 
sents the pole of the ecliptic, and 
MN the path of the pole of the 
equator around it. This vibratory 
motion is called nutation. It is 
principally due to the unequal ac- 
tion of the moon upon the equa- 
torial ring. 

The moon's action, at any given 
time, tends to revolve the ring into 
the plane of its orbit. But, on 
account of the retrograde motion of 
its nodes, the angle between the 
ring and the moon's orbit varies * 

from 1 8J° to 28^-°, going through all the changes every nine- 
teen years. Owing to these changes of position, the equinoxes 
vrill recede sometimes faster, and sometimes slower ; while the 
inclination of the equator to the ecliptic will also increase and 
decrease, causing the poles of the equator to oscillate, a? stated 




ABERRATION OF LIGHT. 85 

above. The amount, by which the pole of the equatoi moves 
to and from the pole of the ecliptic is IS". 

The waves in the figure are exceedingly exaggerated The 
arc MN being about T V of the circumference, the waves, if 
truly represented, would be small enough to cross the arc 270 
times. 

143. Aberration of light. — The heavenly bodies suffer a 
minute apparent displacement, on account of the progressive 
motion of light, combined with the earth's motion in its orbit. 
Suppose the earth to move from C to E (Fig. 
47), while the light, coming from S, describes 
the line I)E. If they arrive together at the 
point E, the impulse on the retina of the eye 
will not be in the same direction as if the 
observer had been at rest ; but the light will 
appear to come in the direction S'E, the body 
being apparently thrown forward from S to T , 
S'. For, make EA = DE, and complete the 
parallelogram CA ; and suppose, according 
to the principle of equal action and reaction, 
that the light has the motion EC given to it, 
in place of the earth's motion, CE ; then the 

two motions, EA and EC, will produce the resultant, EB, as 
though the light had come from S' instead of S. 

144. Aberration illustrated. — The apparent direction of 
any kind of impulse is modified in the same way, by the 
motion of the person who receives it. For instance, if the 
wind drives drops of rain in a person's face, at a certain inclina- 
tion, while he is standing still, when he comes to move toward 
the wind, they will strike him at a less inclination with the 
horizon, as though the source of the drops was further forward. 
For, when the person moves, the effect is the same as if he 
remained at rest, and the wind were to receive an increment 
of velocity equal to his motion. 

145. Greatest and least aberration. — The greatest aberra- 
tion occurs when the body, from which the lis;ht comes, is in a 




86 ADVANCE OF APSIDES. 

direction at right angles to the line of the earth's motion 
The displacement is then 20". 5. When the earth is moving 
directly toward or directly from the body, the aberration is 
zero. Therefore, a star in the plane of the ecliptic is seen in its 
true place once every six months ; bnt three months before 
and three months after either of those times, it is displaced 
20".5 in opposite directions, making the total arc of displace- 
ment 41". But a star at the pole of the ecliptic, being always 
thrown forward of its true place by 20". 5, will seem to de- 
scribe each year a circle, whose diameter is 41". Between the 
ecliptic and its poles, the apparent orbit of aberration is an 
ellipse, whose major axis is 41", and whose minor axis increases 
with the latitude of the body. 

146. Velocity of light computed by aberration. — In the 
triangle AEB (Fig. 47), AB represents the velocity of the 
earth, AEB the observed aberration, and EAB the angle 
between the line of the earth's motion and the direction of 
light. When EAB=90°, the aberration is found to be 20".4451. 
Therefore, 

tan 20".4451 : rad : : 18.393 miles : 185,600 
miles per second, which is about the velocity of light. 

147. Advance of the apsides of the earth's orbit. — It was 
intimated in Art. 74 that the line of apsides is not stationary. 
If the exact place of the perihelion among the stars be noted, 
it will be found the next year 11 ".5 further east — that is, the 
apsides advance 11". 5 per year. But in longitude, the advance- 
is much faster, since the vernal equinox, from which longitude 
is reckoned, retrogrades 50^" per year. The perihelion, there- 
fore, increases its longitude nearly 62" each year. 

As the longitude of the perihelion in 1800 was 279° 30' 8" 
(that is, 9° 30' 8" past the winter solstice) it must have been 
just at the solstice in the year 1247. For, 9° 30' 8" -f- 61f " = 
553 years; and 1800 — 553 = 1247. In a similar manner, it 
is found that the perihelion will be at the summer solstice in 
the year 11741. In the course of many centuries, the length 
and temperature of the seasons are modified by these slo^v 
movements of the equinoxes and the apsides (Art. 75). 



LONGITUDE OF THE SUN. 87 

1 48. Cause of the advance of apsides. — The apsides of the 
earth's orbit are made to advance by the attraction of the 
heavy planets, whose orbits are outside of it. The entire re- 
sultant of the attractions of these planets upon the earth, is to 
diminish a little the earth's tendency to the sun. Hence, as 
the earth approaches one of its apsides, its path is not suffi- 
ciently drawn in by the sun to meet the former line of apsides 
at right angles. But it makes right angles with a radius vec- 
tor a little further on, which becomes, therefore, the new line 
of apsides. 

1 49. Sun's anomaly. — The sun's longitude is his distance 
eastward on the ecliptic from the vernal equinox (Art. 15). 
Its anomaly is its distance eastward, on the ecliptic, from 
perihelion. The reason for reckoning motion from the peri- 
helion is, that the angular velocity depends on it ; so that, to 
find the true longitude of the sun at any time, we need to 
know how far it is from the perihelion. 

150. How to find the true longitude of the sun at a given 
time. — It is first supposed that the sun moves uniformly in a 
circle. And by knowing what its mean motion is, and how 
long it is since it passed the vernal equinox, we have its mean 
longitude at once. But this needs correction on account of 




the variable motion in the ellipse. Let E (Fig. 48) be the 
earth ; PCA, the elliptic orbit of the sun ; and BCF, the sup- 



8S THE MOuN'S DISTANCE. 

posed circular orbit whose area equals that of PCA. Suppose 
the sun's mean place to be at S', and the vernal equinox at °P ; 
then its mean longitude is TDS', already obtained. The angle 
BES' is its mean anomaly. But as the .sun has been passing 
through the nearest part of its orbit, its true place is further 
advanced, as at S. The angle PES is the true anomaly, and 
the difference between them — that is, S'ES — is called the equa- 
tion of the center. This equation, or correction, being found 
in tables of the sun's motions, and applied to the mean longi- 
tude, gives the true longitude. 

If the mean and true places are considered as agreeing at P, 
then the equation of the center immediately becomes positive, 
and increases to its maximum at C ; after which it diminishes, 
and the mean and true places agree again at A. After that, 
the sun falls behind its mean place, and the equation is neg- 
ative, till the sun reaches P, the greatest value being at D. 

The eccentricity of the earth's orbit is so small, that the sun's 
mean and true places never differ so much as 2°, the greatest 
equation of the center being 1° 55' 27". 

151. The anomalistic year. — The perihelion is another 
point from which to measure the revolution about the sun. 
The time of passing round from perihelion to perihelion again 
is called the anomalistic year. It is 4m. 40s. longer than the 
Bidereal year, or 365 d. 6h. 13m. 49s. 



CHAPTEE X. 

T^¥ MOON. — ITS REVOLUTIONS. — ITS PHASES- -THE 
CONDITION OF ITS SURFACE. 

1 52. Distance and dimensions of the moon. — The moon is 
a satellite of the earth, revolving about it within a compara- 
tively small distance, and accompanying it in its orbit around 
the sun. The mean horizontal parallax of the moon at the 



MONTHS. 89- 

earth's equator being 57' 2". 7, its mean distance is found by 
the proportion (Fig. 4), 

sin 57' 2". 7 : rad : : 3962.8 : 238,820m. 

The moon's angular diameter is 31' 6"; therefore, rad : sin 
15' 33" : : 238,820 : 1080.3 ; which is the moon's semi-diameter 
in miles. Hence, the moon's diameter is 2,160.6 miles. 

The surfaces of the earth and moon being as the squares of 
their radii, are as 13 : 1. 

The volumes of the earth and moon being as the cubes of 
their radii, are as 4:9 : 1, nearly. But the moon's density is so> 
small (3.4), that the masses are nearly as 81 : 1. 

The force of gravity on the earth to that on the moon is as- 

W5Voow ::6:1 ' nearly - 

153. Revolution about the earth. — The slightest attention, 
to the position of the moon, from night to night, shows that it 
moves eastward, among the stars, several degrees every day. 
If the instruments of the observatory be employed to measure 
its right ascension and declination, as in the case of the sun 
(Arts. 58, 59), it is ascertained that the moon describes nearly 
a great circle, inclined about 5° to the ecliptic, and occupies- 
27.32 days in returning to the same place among the stars. 

The inclination of the moon's orbit to the ecliptic va~ 
ries from 5° 20' 6" to 4° 57' 22" ; but its mean value is 5° 
8' M". 

154. Months. — The period just mentioned, in which the- 
moon makes a revolution from a star to the same star again, is 
called the sidereal month. The time occupied in making a 
revolution relatively to the sun, instead of a star, is called a 
synodical month. This is more than two days longer than the 
sidereal month ; for the moon's daily progress is about 13° ; 
and during the 27 days of its revolution, the sun, at the rate of 
1° per day, will advance 27°, requiring more than two addi* 
tional days for the moon to overtake it. 

The mean length of the synodical month is 29.53 days. 



90 moon's orbit. 

155. Node*. — The points where the moon's }ath cats the 
circle of the ecliptic are called the moon's nodes. The ascend- 
ing node is the one through which the moon passes from the 
south to the north side of the ecliptic ; the other, 180° from it, 
is called the descending node. 

156. The moon? 8 positions in relation to the sun. — The 
moon is said to be in conjunction with the sun, when both 
bodies have the same longitude ; in opposition, when their 
longitudes diifer by 180°. The conjunction and opposition 
are called by the common name of syzygies. 

When the longitude of the moon is 90°, or 270° greater 
than that of the sun, it is said to be in quadrature. 

The points midway between syzygies and quadratures are 
called octants. 

The period in which the moon passes from any one of these 
points to the same point again — that is, a synodical month — is 
also called a lunation. 

157. To find the synodical month. — The synodical month 
is best obtained by comparing ancient and modern eclipses. 
An eclipse of the sun takes place at the time of conjunction. 
If then, the whole interval between the recorded date of a 
solar eclipse, which occurred before the Christian era, and the 
time of another, which occurred recently, be divided by the 
number of intervening lunations, the quotient is a very accu- 
rate expression of the mean synodical month. 

The mean synodical month, as thus obtained, is 29d. 12h. 
44m. 3s. = 29.5306 days. 

158. To find the sidereal month. — Dividing 360° by 
365.25635, the number of days in a sidereal year, we have 
0°.9856, the mean daily progress of the sun. Multiplying this 
by 29.53, the number of days in a synodical month, we find 
29°.105, the arc passed over by the sun in that time. Now, 
the moon passes over 360° + 29°.105 in a synodical month, 
but only 360° in a sidereal month. Hence, we have the pro- 
portion, 360° + 29°.105 : 360° :: 29.53d. : 2T.32d. 

The sidereal month, more exactly, is 27d. 7h. 43m. lis. 



LIBRATION TN LONGITUDE. 91 

159. Form of the moon's orbit. — It is ascertained by the 
<jame method as was described (Art. 71), that the moon's orbit 
is an ellipse, one of whose foci is at the eirth. The moon's 
apparent diameter varies from 33 f 31" to 29' 21". Therefore, 
the greatest and least distances of the moon from the earth are 
in the ratio of these numbers, or as 8 : 7, nearly ; and the ec- 
centricity = ^ or 0.067, which is about four times as great as 
the eccentricity of the earth's orbit (Art. 73). Yet a figure in 
the exact form of the moon's orbit could not be distinguished 
from a circle, since the major axis would exceed the minor by 
less than T qVo of its length. 

The point of the moon's orbit nearest the earth is called the 
perigee, the most distant point the apogee. 

160. The moon's diurnal motion. — The moon not only re- 
volves about the earth, but also on its own axis in the same 
length of time — that is, once in 27.32 days ; and its axis is 
nearly perpendicular to the plane of its orbit. This rotation 
is indicated by the fact that the same side of the moon is al- 
ways presented toward the earth. If it should pass around the 
earth, and not turn upon an axis, it would obviously present 
all sides to us in the course of each revolution. 

But though it keeps the same side toward the earth, it pre- 
sents all sides to the sun once in each synodical month ; there- 
fore, the days and nights on the moon are nearly 30 (29.53) 
times the length of those on the earth. 

101. The moon's librations. — Though the same side of the 
moon is turned toward us on the whole, yet there are slight 
apparent oscillations, by which narrow portions of the other 
hemisphere alternately come into view. These are called 
librations. They are of three kinds : the libration in longi- 
tude, the libration in latitude, and the diurnal libration. 

162. The libration in longitude. — By this libration we ex- 
tend our view a little further round upon the moon's equator, 
first on one side, then on the other, every sidereal month. 

It arises from the fact that while the moon rotates uni- 
formly on its axis, it revolves in its elliptical oibit with un- 



92 REVOLUTION" ABOUT THE SUN. 

equal angular velocity. Near the apogee, where it moves 
slowest, it rotates more than 90° on its axis, while passing just 
90° around us, and thus reveals a little of the remote hemi- 
sphere on the eastern side. Near the perigee, on the other 
hand, where the orbit motion is rapid, it makes less than one- 
fourth of a rotation, while going 90° around the earth. This 
brings into view a little of the other hemisphere on the western 
limb. 

If the moon's orbit were a circle, there would be no libra- 
tion of longitude. 

163. The libration in latitude. — As the name implies, this 
libration extends our view alternately north and south on the 
moon's meridian. As the moon's equator is a little inclined 
to the plane of its orbit, its north and south poles are brought 
alternately toward us, just as the earth's poles are presented in 
turn toward the sun every year. The mean value of the incli- 
nation of the moon's equator to its orbit is 6° 39'. 

If the moon's equator and its orbit were in the same plant, 
there would be no libration of latitude. 

164. The diurnal libration. — This is the effect of diurnal 
parallax. When the moon is on the meridian, we view it 
nearly as from the center of the earth ; but when it is at the 
horizon, we see it, as it were, from a position near 4,000 miles 
higher, and extend our vision a little distance over its western 
limb at rising, and its eastern at setting. 

165. Apparent diameter on the meridian and at the hori- 
zon. — The distance of the moon from the earth is about 60 
times the radius of the earth. Therefore, when the moon is on 
the meridian, as it is ^ nearer than when at the horizon, its 
apparent diameter is -^ greater. This change, equal to about 
30", is too small to be perceived by the eye, but can be meas- 
ured by instruments. 

166. The moon's revolution about the sun. — While the 
moon revolves about the earth, the earth revolves about the 
sun, at a distance 387 times as great. For, 238,820 x 387 = 
92, 423,0u0. 



WHAT FOECES CONTROL THE MOON. 93 

Therefore, the moon really has a third revolution — namely, 
that in company with the earth around the sun. And this is 
far greater than its other revolutions, which have been de- 
scribed. A point of the moon's equator, in its diurnal motion, 
£oes only 10 miles per hour. Around the earth, the moon's 
velocity is nearly 2,300 miles per hour ; but around the sun, it 
is more than 66,000 miles per hour. 

167. Form of path around the sun. — "Whenever a body re- 
volves about a center, while that center is itself in motion, the 
body describes a species of curve, called an epicycloid. The 
moon's path about the sun is a leaving epicycloid. Let the 
small circles at A, B, etc. (Fig. 49), represent the size of the. 

Fig. 49. 




moon's orbit, and let AE be an arc of the earth's orbit, the 
sun being at the intersection of the dotted lines when pro- 
duced. While the moon describes one half of its orbit, the 
earth goes over ^j of its annual circuit — that is, from A to E. 
Therefore, the earth being at A, suppose the moon in quadra- 
ture on the left, beginning to describe the semicircle nearest the 
sun. When the earth reaches B, the moon has passed to the 
octant m / at C, the moon is in conjunction ; at D, it is at the 
next octant ; and at E, it is again in quadrature on the right, 
having described a semicircle relatively to the earth. But, in 
relation to the sun, it has passed over the curve inside of the 
earth's path, from Ammm'E. At E, it crosses the earth's path, 
and while describing the outer semicircle, it advances with the 
earth a distance equal to AE, on the outside. Thus, the 
moon's path around the sun consists of 25 undulations, so 
Blight that, if represented alone, the whole would scarcely be 
distinguished from the earth's orbit. 

168. By what forces the moon is mainly controlled. — Since 
the moon describes around the sun an orbit at the mean d;s- 



94 moon's phases. 

tance of the earth's orbit, and in the same time, it must be 
subject to the same projectile and centripetal forces. It the 
earth, therefore, were to be annihilated, the moon's path about 
the sun would not be essentially disturbed ; the waves only 
would cease, and the orbit become an exact ellipse. 

The relative attractions of the earth and sun, exerted on 
the moon, are estimated by the formula proved in Art. 92, 

c oo — . Considering the radius of the moon's orbit = 1, that 

of the earth's orbit is about 387; and the times are 27.32d. and 
365. 25d., respectively. Hence, attraction to the earth : that 

i ^87 

to the sun : : 3 : pgijgy : : ] : 2 ' 2 ' near1 ^ Therefore, 

the sun, though so very far frpm the moon, exerts upon it 
2 J times more attraction than the earth does. 

169. How the earth's action causes the waves in the moon's 
path. — When the moon is in conjunction, as at C, the earth 
draws it away from the sun, so that it begins to move further 
off, as at D, E, etc., till it reaches opposition. But, at opposi- 
tion, the earth is on the same side as the sun, and increases the 
moon's tendency toward it, so that the moon begins to move 
toward the sun, and continues approaching till it reaches con- 
junction again. But, in describing the wave line, the moon 
sometimes gets in advance of the earth in its orbit, as at A, 
and then falls behind, as at E. For, the earth at A draws the 
moon backward, and it falls further and further back, till it is 
behind the earth in its motion, as at E, where the earth, having 
overcome the backward motion, draws it forward, till it passes 
by, and is again in advance of the earth. Thus, in the moon's 
great revolution around the sun, we may regard its path as 
thrown into the waving line by the small disturbing influences 
of the earth. 

170. Phases of the moon. — The moon is not self-luminous, 
and is seen only as it reflects to us the light which falls upon 
it. The several forms which the part illuminated by the sun 
presents to our view, are called phases. 

The circle of illumination, or the terminator, is the circle 



MOON'S PHASES. 



95 



which separates the hemisphere enlightened by the sun from 
the dark hemisphere, and is perpendicular to the sun's rays 
which fall on the moon. The circle of the disk is that which 
separates the hemisphere turned toward the earth from the op- 
posite one, and is perpendicular to our line of vision. The 
phase depends on the size of the angle formed at the moon, 
between the solar ray and our visual line. 

Fig. 50. 



O 




Let the earth be at E (Fig. 50), and the moon in several po- 
sitions, A, B, etc., and let the lines AS, BS, etc., be directed 
toward the sun. At A, the moon is in conjunction, and wholly 
invisible — this is called new moon; and the angle SAE, be- 
tween the solar ray and visual ray, is 180°. From A to 
(as at B), the phase is called crescent ; and the angle, SBE, is 
obtuse. The first quarter occurs at C, the quadrature, where 
SCE is a right angle. From C to F (as at D), the phase is 
called gibbous ; in this phase, the angle, SDE, is always acute. 
At F, the moon is in opposition, and wholly illuminated. This 
is called full moon ; the angle, SFE, is 0°. From F to A, 
the phases are repeated in reverse order, the last quarter being 
at H. The outer figures at B, C, etc., show the corresponding 
phase 



t)6 INEQUALITIES OF THE MOON'S SURFACE. 

171. The meridian altitudes of the moon at a given phase. 
— It is generally observed that at a given age of the moon, foi 
instance at the full, its meridian altitude is very different at 
different seasons of the year. This is readily explained, by 
noticing the moon's relations to the sun. As the moon's path 
is everywhere near the ecliptic, the new moon will culminate 
at a high point when the sun does — that is, in the summer. 
But, in the same season, the fall moon, being opposite to the 
sun, will culminate low. On the contrary, when the sun is in 
the most southern part of the ecliptic, and culminates low, as 
is the case in winter, the new moon will do so likewise ; but 
the full moon will culminate at a high point. In the polar 
winter, therefore, wdien the sun is absent for months, the moon, 
whenever near the full, circulates round the sky without 
setting. 

1 T 2. The harvest moon. — This name is given to the full 
moon which occurs nearest to the autumnal equinox, Septem- 
ber 2 2d, and which rises from evening to evening with a less in- 
terval of time than the full moon of any other season. 

The sun being at the autumnal equinox, the moon is near 
the vernal equinox, and at sunset, the southern half of the 
ecliptic is above the horizon, and makes the smallest possible 
angle with it. It is this small angle, made by the ecliptic, and 
therefore by the moon's orbit with the horizon, which causes 
the small interval in the time of the moon's rising from one 
evening to another; for, as the moon advances 13° each day 
in its orbit, this arc is so oblique to the horizon that its two 
extremities rise with only a few minutes' difference of time ; 
but the place of rising moves rapidly northward. 

The harvest moon attracts most attention in high latitudes, 
where the angle between the ecliptic and horizon is smaller, 
and therefore the intervals of time are less. 

The moon passes the vernal equinox every month, and 
therefore rises with the same small intervals. But when the 
moon is not full at the same time, the circumstance is un- 
noticed. 

173. Inequalities of the moon's surface. — These are clearly 



FORMS OF VALLEYS. 97 

revealed by the changing direction of the sun's rays. As 
the terminator advances over the disk, the light strikes the 
highest peaks, which appear as bright points a little way upon 
the dark part of the moon. After the terminator has passed 
over them, they project shadows away from the sun, which 
correspond to the apparent shape of the elevations, and grow 
shorter as the rays fall more nearly vertical. And again, in 
the waning of the moon, the shadows are cast in the opposite 
direction, lengthening until the dark part of the disk reaches 
them, and the summits once more become isolated bright 
points, and then disappear. Fig. 2, Frontispiece, will give 
some idea of these appearances. 

174. Forms of valleys. — The most striking characteristic 
of the moon's surface is its numerous circular valleys. A few 
are represented in Figs. 1 and 2, Fr. The smaller and more 
regular ones are of all sizes, from one or two miles in diameter 
up to sixty miles. These are numbered by hundreds. The 
mountain ridge which surrounds one of these cavities is a ring, 
very steep and precipitous on the inner side ; but externally it 
falls off by a rugged but gradual slope. These ridges are 
called ring-mountains. In the central part of the cavity are 
generally one or more steep, conical mountains. Some of the 
principal ring-mountains are No. 1. Tycho; 2. Kepler; 3. Co- 
pernicus ; etc. (Fig. 1, Fr.) 

There is another class of larger but less regular cavities, 
sometimes called bulwark plains. Their diameters are often 
more than one hundred miles. These are also surround- 
ed bj rough mountain masses arranged in a circle. Over 
these plains are sparsely scattered small conical and ring moun- 
tains. 

There are still larger tracts, more level than the general 
lunar surface, and of a darkish hue, which still retain the name 
of seas, formerly given them, though they are covered with 
permanent inequalities, and show no signs of being fluid. Ex- 
amples of these are: A, mare humorum; B, mare nubium^ 
ntc (Fig. 1, Fr.) 

Besides the ridges of mountains inclosing the circular val- 

7 



98 LUNAR MOUNTAINS. 

leys, there are occasional chains and spurs, having more resem- 
blance to terrestrial ranges.* 

175. Luminous radiations. — At fall moon, all shadows 
disappear, because the light falls in the direction of our line of 
vision. But at that time another peculiarity presents itself. 
From a few of the large ring mountains there radiate a great 
number of luminous stripes, nearly in straight lines, and ex 
tending, in some cases, hundreds of miles. They are not 
ridges, as they cast no shadows when the terminator passes 
them ; and the difference of illumination must result from the 
different nature of their material. They are sometimes called 
lava-lines. The most extensive system occurs around Tycho, 
marked 1, in Fig. 1, Fr. 

176. Surface rigid and angular. — Every part of the moon's 
surface has the appearance of rocky hardness. The interior 
slopes of the ring-mountains are steep, rough, and angular. 
The conical peaks within them appear like isolated rocks, re- 
sembling the needles of the Alps. The surface nowhere gives 
indication of having been softened down by the action oi 
water. 

17 7. Probable volcanic origin. — The circular cavities, with 
steep and rugged sides, appear like vast craters, and the moun- 
tains within them like volcanic cones, more recently thrown 
up. Nearly every part of the hemisphere presented to our 
view exhibits these indications of former volcanic action, on a 
scale far beyond any thing on the earth. But there is no evi- 
dence of volcanic action at present. 

178. Height of lunar mountains. — One method of measur- 
ing the height of a lunar mountain is the following. Let the 
light from the sun, S (Fig. 51), pass the moon's surface at O, and 
illuminate the summit of the mountain, MF. To the observer 
on the earth at E, M is seen as a bright point beyond the ter- 

* The lunar map of Beer and Madler, 2$ feet in diameter, contains a very 
perfect delineation of the mountains and valleys of the moon, accompanied 
bv their names. 



NO ATMOSPHERE OR VAPOR. 



90 



minator O. Let OEM, subtended by OM, be measured with 
a micrometer; also, OEB, between the terminator and the 
opposite edge of the disk. From the latter subtract CEB, the 
semi-diameter, and OEC is known ; which, with OC and EC, 
will give EO and EOC. The supplement EOA, plus 90°, 
equals EOM. Then EO and the angles E and O, will furnish 
OM ; from which and 00, CM is computed. FC, subtracted 
from CM, leaves FM, the height required. 

Fig. 51. 




The height of a mountain may also be determined by meas- 
uring the length of its shadow, and the inclination of the solar 
ray which casts it. 

The highest of the lunar mountains have an elevation of 4^ 
miles, and great numbers of them exceed three miles. Thus,. 
the mountains of the moon are proportionally much greater 
than those of the earth. For, while the diameter of the mooD 
is not much more than one-fourth as great as the earth's diam 
eter, its mountains are about equal in height to the mountains 
on the earth. 



179. No atmosphere or vapor. — If any kind of atmosphere 
were spread over the disk of the moon, it would reflect the 
sun's light so strongly as to dim the features of the solid sur- 
face. Nothing of the kind is ever perceived. No terrestrial 



100 NO ATMOSPHERE OR VAPOR. 

objects, however near, ever exhibit greater sharpness of outline 
than the inequalities of the moon ; and they never vary in this 
respect, except in a manner which is obviously occasioned by 
our own atmosphere. 

But the severest test of a perceptible atmosphere would be 
the effect on a star at the beginning and end of its occultation 
by the moon. Let AB (Fig. 52) be the edge of the moon's 

Fig. 52. 




disk, and CD that of the atmosphere around it. The light 
from the star S will, according to the laws of optics, be re- 
fracted toward the moon in entering its atmosphere, and as 
much more in the same direction in leaving it ; so that it will 
reach the observer at E, appearing to come from S', when the 
star is really behind the moon at S. Thus, it will appear to be 
detained in its diurnal motion as it approaches the edge of the 
moon, and to arrive only to S' when it has really reached the 
position S. So, also, in reappearing at the opposite limb, the 
star will seem to have advanced to the edge, when it is 
still behind the moon ; so that, after coming into view, and 
before passing by the atmosphere, it will again appear to be 
detained in its diurnal motion. Since it disappears too late, 
and reappears too early, the duration of occultation is too 
short. 

Besides this irregularity in its motion, its brightness will 
also be a little dimmed by the obstruction of the atmosphere, 
just before disappearing, and just after reappearing. 

]STow, the nicest observations have failed to show either oi 
these effects. The diurnal motion is uniform up to the very 
edge of the disk, and the actual continuance of occultation is 
equal to the calculated duration. And, as to loss of light, 
the star at its full brightness disappears all at once, with a 
suddenness which is startling. Its reappearance is equally 
sudden, and without any change of intensity in its light. The 
moon, therefore, has no appreciable atmosphere. 



VIEW OF EARTH FROM MOOJN T . 101 

180. Changes of temperature on the moon. — The moon's 
equator makes an angle of only 1J° with the ecliptic, and 
therefore experiences no perceptible change of seasons ; but its 
diurnal rotation is so slow, that the extremes of heat and cold 
during each day are excessive. A place on the moon is ex- 
posed to the full power of the sun's rays for about two weeks, 
and then is for as long a time turned away from the sun, with- 
out clouds, or even air, to prevent the free radiation of heat. 

181. View of the earth from the moon. — 

1. As to magnitude. — The apparent dimensions of the two 
bodies, as seen one from the other, are proportional to their 
real dimensions. Hence, in diameter, the earth as seen from 
the moon is 3} times as large as the moon viewed from the 
earth, and in area is about 13 times as large. 

2. As to phase. — It is obvious, from Fig. 50, that when the 
full moon is presented to the earth, the earth's dark side is 
toward the moon, and the reverse. Also, that when we see 
the gibbous phases of the moon, a spectator on the moon would 
eee crescent phases of the earth ; for the angle SED or SEG 
would then be obtuse. In like manner, the relative phases are 
in every case supplementary to each other. This relation ex- 
plains the w T ell-known fact that near the time of new moon, 
the part of the moon not directly enlightened by the sun is 
distinctly visible. It is then illuminated indirectly by the 
earth, which is nearly full as seen from the moon, and reflects 
a strong light upon it. 

For the same reason, the moon can be faintly seen in a total 
solar eclipse. 

3. As to position in the sky. — The earth seen from the moon 
has no apparent diurnal rotation, as all other heavenly bodies 
have, but remains nearly fixed in the same part of the sky. 
This is owing to the fact that the moon's monthly motion and 
its diurnal motion are at the same rate in the same direction, 
so that one apparent motion of the earth neutralizes the other. 
Hence, a spectator occupying the middle of the moon's disk 
sees the earth perpetually near his zenith. Another, at the 
edge of the disk, sees it always near the same point of the 
horizon. 



102 moon's gravity disturbed. 

The first ana second librations of the moon, since they vary 
the spectator's position a little in relation to the disk, merely 
canse small oscillations of the earth's place in the sky. 

4. As to surface -The earth, by its rotation, presents all its 
parts to the view of the nearer hemisphere of the moon once in 
25 hoars. To the other hemisphere it never appears at all. 

On account of its nearness, and its great size, we might sup- 
pose that the geographical features of the earth would be 
very conspicuous to a spectator on the moon, and that the 
nature of its surface in nearly all respects could be thoroughly 
observed. But the deep and dense atmosphere of the earth 
would reflect an intense light, so as probably to render the in- 
equalities of the terrestrial surface nearly invisible ; and when- 
ever clouds prevail over a country, that portion of the earths 
surface would, of course, be entirely hidden from view. 



CHAPTER XI. 

DISTURBANCES OF THE MOON'S MOTION CAUSED BY THE SUN.. 

182. Why the sun disturbs the moon's revolutions around 
the earth. — If the sun were at an infinite distance from the 
earth and moon, however great its attraction might be, it 
would not disturb their mutual relations, because it would act 
on both exactly alike. Though the sun's distance from them 
is very great, being 387 times their distance from each other, 
yet the difference of action is sufficient to produce sensible dis- 
turbances. These disturbances are caused in part by difference 
of distance, and in part by difference of direction, 

183. The moon's gravity diminished at syzygies, and in- 
creased, at quadratures. — When the moon is in conjunction, 
the sun attracts it more than it does the earth, in the ratio of 
387 2 : 386 2 , and thus diminishes the moon's tendency to the 
Barth. In opposition, the sun attracts the moon less than it 
Joes the earth in nearly the same ratio, which, as before, di- 



THE SUN S DISTUEBING EFFECT. 



103 



smnishes the moon's tendency to the earth. Therefore, at the 
syzygies, the moon's gravity to the earth is diminished. And 
the diminution is computed to be about g\ of the whole. 

In quadrature, the sun attracts the moon in a line slightly 
oblique to that in which it attracts the earth. Hence, there is 
a small component of its action directed toward the earth. 
Therefore, at the quadratures, the moon's gravity to the earth 
is increased. This increase is proved to be about t -1q of the 
whole, or one-half as great as the diminution at syzygies. 

As the diminution at syzygies is more than the increase at 
quadratures, the entire effect of the sun's influence is to dimin- 
ish the moon's gravity to the earth, and thus cause it to revolve 
in a larger orbit than it would do if the sun did not exist. 
The moon's gravity to the earth is diminished by 3-J0, in con- 
sequence of the sun's action. 



184. The sun's disturbing effect repre- 
sented geometrically. — Let ABCD (Fig. 
53) be the moon's orbit described about 
the earth E, and S the place of the sun. 
Suppose the moon at M. Let ES rep- 
resent the attraction of the sun upon the 
earth. Then (Art. 128), SM 2 : SE 2 : : SE : 
SF 3 
^rp = the attraction of the sun upon M, 

in the direction MS. Make MG = £==, 

SM 2 ' 

draw MF equal and parallel to ES, and 
complete the parallelogram MFGH. 
Eesolve the force MG- into MF and MH. 
Since the component MF is equal and 
parallel to ES, which is the sun's attrac- 
tion on the earth, it produces no disturb- 
ance ; and the only force which can dis- 
turb the relations of M and E is the 
other component MH. This line lies 
in various positions, and is of various 
lengths, according to the place of M. It 
is convenient to reduce it to two ele- 




104 



EVECTION. 



ments called the radial and the tangential disturbing forces 
Draw MO tangent to the orbit, and EM joining the earth 
and moon ; then, ME may be resolved into the radial force 
MP, increasing or diminishing the moon's gravity to the earth, 
and the tangential force MO, which increases or diminishes 
the velocity of the moon. In the figure, the position of MH 
is such, that MP increases the gravity, and MO accelerates. 

Near the quadratures, MP acts toward E ; and near the 
syzygies, it acts away from E. MO accelerates on the quad- 
rants DA and BC, and retards on AB and CD. 

185. Equations for correcting the moorts place. — The 
moon's path being elliptical, and its motion being subject to 
several disturbances, its true longitude for a given time can 
not be found, except by applying various corrections. 

186. TJie equation of the center, — First suppose the moon 
to revolve uniformly in a circular orbit, and then, as in the 
case of the sun (Art. 150), apply the equation of the center to 
change its place for the variable motion in the ellipse. The 
moon's orbit being more eccentric than the earth's, its great- 
est equation of the center is 6° 18' 17", while the sun's is less 
than 2°. 



187. Evection. — A cor- 
rection must be applied on 
account of the change of ec- 
centricity caused by the 
sun's disturbance. This 
change of eccentricity is 
called evection. It is caused 
by the radial disturbance 
MP (Fig. 53), which pro- 
duces greater or less effect, 
according to the position of 
the line of apsides in re- 
lation to the line of syzy- 
gies. Let FH (Fig. 54) 
be the line of apsides of the moon's orbit about the earth, 




VARIATION. 105 

E, and suppose the sun to be in the direction A. Then AC 
is the line of syzygies, and the two lines coincide. The 
moon's gravity toward E is diminished at F and H, as it always 
is when in the line of syzygies. But at F, it is diminished less 
than ever, because there is the least difference of distances, AE 
and AF ; while at H, it is diminished more than ever, because 
the difference of distances AE and AH is the greatest possible. 
Hence, F is less separated from E, and H more separated from 
E, than in any other situation. The same would be true, if the 
sun were in the direction of C. Therefore, when the line of 
apsides coincides with the line of syzygies, the moon's orbit is 
most eccentric. 

Again, suppose the sun to be in the direction B or D ; in 
other words, that the line of apsides is in quadrature. Then, 
the gravity of the moon toward E is increased at F and H, as 
it always is when in quadrature. But at F, its increase is the 
least possible, because the obliquity of FB to EB is the least 
possible ; while at H, the increase is the greatest, because the 
obliquity of HB to EB is the greatest. Hence, HE is less, 
compared with FE, than in any other position. Therefore, the 
eccentricity is least when the line of apsides is in quadrature. 
The greatest correction for evection is 1° 12'. 

188. Variation. — Another correction is applied on account 
of the alternate changes of velocity caused by the sun. This 
change of velocity is called variation. It is produced by the 
tangential disturbance MO (Fig. 53). From D to A, it con- 
spires with the motion of the moon, and accelerates it. From 
A to B, it is directed backward, and retards the moon's motion. 
From B to C it accelerates, and from C to D it retards. It 
might be supposed that because the sun attracts toward S, this 
would act against the moon's motion in going from B to 0, 
and thus retard it ; and with it from C to D, and accelerate it. 
But the disturbing action is not the absolute, but the relative 
attraction. From B to C, the sun attracts the moon less than 
it does the earth ; and the effect is the same as if it exerted no 
attraction on the earth, and urged the moon in the opposite 
direction — that is, toward C. Hence, the moon's velocity is 
alternately accelerated and retarded in the successive quad 



i06 KETKOGRADATION OF THE MOON'S NODES. 

ants, causing the greatest equation about 45° from the quad« 
ratures B and D. The variation at its maximum is about 37'. 

189. Annual equation. — This is a change in the moon's 
motion, arising from the greater and less distance of the sun at 
different seasons of the year. The disturbing action of the sun 
is greatest when it is nearest — that is, at perihelion ; and it is 
least when it is most distant, or at aphelion. This inequality 
is called the annual equation, since it passes through all its 
changes in a year. It amounts to about 11'. 

190. Smaller equations. — The foregoing are the largest in- 
equalities of the moon's motion, which require corrections to 
be made for finding its true place. There is a large number of 
smaller ones, for which allowance must be made, in order to 
obtain the moon's longitude for a given time, with the utmost 
exactness. By the most complete tables of the moon now in 
use, its place can be determined within 3". 

191. Apsides of the moon's orbit. — The line of apsides ad- 
vances — that is, moves forward — from west to east. This is a 
disturbance produced by the sun, and is explained in the same 
manner as the advance of the earth's apsides (Art. 148). The 
attraction of a body external to the orbit always tends to pro- 
duce this effect. Though the sun makes the moon's gravity to 
the earth sometimes greater, and bometimes less, yet it, on the 
whole, diminishes it (Art. 183). Without any disturbing in- 
fluence, the moon would always describe the same elliptic 
orbit. But as it approaches one of its apsides, it is, in general, 
not sufficiently drawn in toward the center to cut the former 
line of apsides at right angles ; but it makes right angles with 
a radius vector a little further on, which, therefore, becomes 
the new line of apsides. The apsides of the earth's orbit ad- 
vance with exceeding slowness (Art. 147); but the sun's dis- 
turbing power is so great, that those of the moon's orbit shift 
their place more than 3° in each sidereal month, and, therefore 
make a complete revolution in about 9 years. 

199. Betrogradation of the moon's nodes. — In the pre- 



RETROGRADATION OF THE MOON'S NODES. 



107 



ceding articles we have considered the disturbing force of the 
sun upon the moon in the plane of its orbit, one component 
being the radial, and the other the tangential. As the moon's 
orbit does not coincide with the ecliptic, the sun exerts another 
disturbing force — namely, out of the moon's orbit either to or 
from the ecliptic. This third force is called the orthogonab 
component. It causes a motion of the nodes, and a change of 
inclination. 

Fig. 55. 




Let MN (Fig. 55) represent a short arc of the ecliptic, AB 
an arc of the moon's orbit, and ANM the inclination of their 
planes. "When the moon is at L, moving toward the node, N, 
the sun attracts it in a line slightly oblique to its orbit. There- 
fore, while one component of this disturbing force lies in the 
plane of the orbit, the other is perpendicular to it. Let 12> be 
the distance through which the latter would move the moon in 
the time of its describing ~La in its orbit. The resultant is Le, 
cutting the ecliptic in W. Again, after passing the node, let the 
orthogonal component move the moon over Ud, while it would 
describe Ue in its own plane. Then, by the joint action of Ue 
and Ud, it describes Uf which produced makes the node at W. 
In the case here described, the line of nodes is supposed to lie 
perpendicular to the line joining the earth and sun, and we see 
that the node is made to move backward, both when the moon 
approaches it, and when it is leaving it. But in other posi- 
tions of the line of nodes, it can be shown that the orthogonal 
component is directed sometimes toward the ecliptic and some- 
times from it. In the former case the nodes retrograde, in the 



108 ACCELERATION OF THfi MOON. 

latter they advance. In any revolution, however, the latter 
effect is less than the former ; so that on the whole, the nodes 
have a retrograde motion. 

The nodes of the moon's orbit retrograde at the rate of 19° 
35' each year, thus completing a revolution in 18.6 years. 

193. Disturbance of the inclination of the moon's orbit. — 
When the moon approaches a node, the inclination of its orbit to 
the ecliptic is generally increased ; for LN'M is greater than the 
interior angle LNM. And it is generally diminished as the 
moon leaves a node, since ZN"1$ is less than the exterior angle 
£N"'!N". These alternate changes nearly balance each other, and 
leave the mean value of the inclination almost constant— 
namely, 5° 8' 44" (Art. 153). 

1 94. Periodical and secular inequalities. — The inequalities 
in the moon's motion, which have been described, pass through 
all their changes in a short period, as a month, a year, or a 
few years at most. These are called periodical. But there are 
others, whose periods extend through many centuries or ages. 
These are called secular. Some minute secular disturbances in 
the solar system run on in the same direction for an indefinite 
number of centuries. 

195. The acceleration of the moon' ] s mean motion. — This is 
an interesting example of secular inequality. The period of a 
lunation is now sensibly shorter than it was before the Chris- 
tian era. This is ascertained by comparing the recorded date 
of an eclipse which occurred in 720 before Christ with the 
time of any recent eclipse. The whole interval, if divided by 
the present mean length of a lunation, leaves a considerable 
remainder. The acceleration amounts to about 10" in a cen- 
tury. 

196. Its cause. — It has been stated that the sun diminishes 
the moon's gravity toward the earth (Art. 183). The amount 
of this diminution depends, in part, on the eccentricity of the 
earth's orbit. From the time of the earliest observations, the 
earth's orbit has been slowly approaching a circle, and will 



ECLIPSE MONTHS. 109 

continue to do so for many centuries to come. So long as the 
eccentricity of the earth's orbit is diminishing, the sun's dis- 
turbing action on the moon diminishes also. The moon, there- 
fore, being less drawn away from the earth, describes a smaller 
orbit, and, consequently, in a shorter time. In the course ot 
ages, the earth's orbit will reach the limit of its change, and 
begin to grow more eccentric. The moon's orbit will then 
commence to enlarge, and will, therefore, require a longer 
time to be described. 



CHAPTER XII. 

ECLIPSES OF THE MOON. — ECLIPSES OF THE SUN. 

197. General relations in eclipses. — The moon is eclipsed, 
when it is obscured wholly or in part by the earth's shadow. 
It can occur, therefore, only at opposition, or full moon. The 
sun is eclipsed, when it is either wholly or partially concealed 
from view by the moon coming between it and the earth. This 
can happen only at conjunction, or new moon. 

If the moon's orbit and the ecliptic were coincident planes, 
there must be an eclipse of the moon at every full moon, and 
an eclipse of the sun at every new moon ; for at those times 
the three bodies would be in a straight line. But as the moon's 
orbit and the ecliptic make an angle of 5° with each other, the 
moon generally passes opposition and conjunction so far north 
or south of the sun, that there is no eclipse. That an eclipse 
may occur, the syzygies must happen near the line of nodes, so 
that, as the moon comes into conjunction or opposition, some 
parts of the three bodies may be in a straight line. 

198. Eclipse months. — As there are two nodes on opposite 
sides of the heavens, the sun in its annual progress must pass 
through both of them every year, at intervals of about six 
months. And as the moon comes into the line of syzygies 
every two weeks, the sun will certainly be near enough to a 



110 earth's shadow. 

node for one or two eclipses, and possibly for three, every six 
months. Thus, the eclipses of any year always occur in clus- 
ters, at opposite seasons. If two or three are in January, the 
others are in July. These are called the node months of that 
year. In 1884, for example, the node months are parts ol 
March and April, and parts of September and October. On 
account of the rei/ograde motion of the nodes, the sun passes 
from a node to the same one again in less than a year, so 
that the node months occur earlier each successive year per- 
petually. 

*199. Eclipse of the moon. — When the moon is eclipsed, 
there is nothing interposed to hide it from our view ; but it 
merely falls into the shadow of the earth, and is obscured. 
This obscuration may possibly continue for several hours. 

200. Form and angle of the earth? s shadow. — As the sun 
is larger than the earth, and both are spheres, the tangents 
drawn from one to the other, along the corresponding edges, 
will converge and form a cone. Thus (Fig. 56), let AA' be 
the sun, and BB' the earth ; then BB'C is the conical shadow ; 
and rays of light from AA', moving in straight lines, can not 
enter any part of it. The axis of the shadow, EC, is the ex- 
tension of the line joining the centers of the sun and earth. 
Since the light is entirely excluded from the cone BB'C, it is 
often called the total shadow. 

Fig. 56 



IToin AE; then the exterior angle AES = ACE + EAC; 
.-. ACE = AES - EAC. But AES is the sun's apparent 
semi-diameter, and EAC is the sun's horizontal parallax. 



LUNAR ECLIPTIC LIMIT. Ill 

Therefore, the semi-angle of the earth's shadow is equal to th6 
sun's apparent semi-diameter diminished by its horizontal 
parallax. Calling the sun's semi-diameter d, and its horizontal 
parallax^?, the semi-angle of the shadow is 6 — p. d = 16' 2", 
and p = 8".8; .\ 6 —p = 15' 53".2, the mean value of the 
semi-angle of the shadow. 

201. Length of the earth's shadow. — In the triangle ECB 
right-angled at B, as we know EB and ECB, EC is found by 
the proportion, sin {p — p) : rad : : 3956 : 856,050, the length 
of the earth's shadow in miles. 

Since the moon is 238,820 miles from the earth, the length 
of the earth's shadow is more than 3 i times the distance from 
the earth to the moon ; and the moon, when eclipsed, passes 
through the broader part of it. 

202. Angular breadth of the section traversed by the moon. 
— Let h'h be a part of the moon's orbit supposed to pass 
through the axis of the shadow at M. Then Mm is the semi- 
diameter of the section, and MEm its angular semi-diameter, 
which is to be found. The exterior angle EmB = ECm + 
CEra ; .\ CEra = EmB — ECm. But EmB is the horizontal 
parallax of the moon, and ECm the semi-angle of the shadow. 
Call the moon's parallax P, then the angular semi-diameter of 
the shadow = P — (6 — p\ or P + p — 6. 

P = 57' 3", and 6 - p = 15' 53" ; .-. P + p - S = 41' 10' , 
the mean semi-diameter of the section. 

Since the moon's semi-diameter is 15 r 33", the breadth of 
shadow where the moon crosses it is 2f times the breadth of 
the moon. 

203. Lunar ecliptic limit. — The distance of the center or 
the earth's shadow from the node, when the moon at opposi- 
tion would only touch the shadow, is called the lunar ecliptic 
limit. Let GN (Fig. 57) be an arc of the ecliptic, MN an arc 
of the moon's orbit, N the node, Ca the semi-diameter of the 
6hadow, and aM that of the moon when it only touches the 
shadow at opposition. Then CK is the ecliptic limit. In the 
spherical triangle CMN, right-angled at M, N being known. 



112 DIMENSIONS OF PENUMBRA. 

and also Qa and f/M, we have, by Napier's rule, rad x sin CM 
= sin CN x sin N ; from which CN is obtained. Since N", 
C«, and aM are all variable, CX must also vary. Its greatest 
value is 12° 24/, beyond which an eclipse is impossible. Its 
least value is 9° 2i r within which an eclipse can not fail to 
*>ccur. 

Fig. 57. 




204. Magnitude of eclipse. — The mere contact of the moon 
and earth's shadow at the ecliptic limit is called an appulse. 
If the moon is obscured only in part, the phenomenon is called 
a partial eclipse. It is a total eclipse when the moon is en- 
tirely enveloped in the shadow. If its center passes through 
the axis of the shadow, there is a central eclipse. 

205. The earth? s penumbra. — If tangents be drawn across 
the opposite sides of the sun and earth, as Ah, A!h (Fig. 56), 
they diverge, and inclose a space around the total shadow, 
called the penumbra, or partial shadow. Its form is the frus- 
tum of a cone, and it extends to an infinite distance beyond 
the earth. Within the penumbra, and outside of the shadow, 
there is light from a part of the sun only, while the other part 
is conceaJed by the earth. Thus, at a point between BC and 
"Bh produced, it is obvious that the limb of the sun near A 
could not shine, because the light would be intercepted by the 
opposite side of the earth near B. The vertex of the penumbra 
is between the earth and sun, at C. 

206. Dimensions of the penumbra. — The semi-angle of the 
penumbra is h'Q'C (Fig. 56), which is equal to AC'S. And the 
external angle AC'S --= EAC + C'EA. But EAC is the sun's 
horizontal parallax = p ; and C'EA is the s tin's apparent semi- 



SOLAR AND LUNAR TABLES. 113 

diameter = d; therefore, the semi-angle of the penumbra = 

The semi-diameter of the section of the penumbra through 
which the moon passes is AM, and its angular or apparent 
semi-diameter is AEM. And AEM, being external to the tri- 
angle AEC, equals EC'A + EAC. But EAC is the moon's 
horizontal parallax = P ; and EC'A = p + d ; therefore, the 
apparent semi-diameter of the earth's penumbra = P -\- p + 6 
At mean values, this equals 1° 13 / 14", which is nearly 5 times 
the semi-diameter of the moon. 

207. Effect of the penumbra. — On account of the penum- 
bra, the edge of the total shadow is not sharply denned, but 
shades off into the full light by slow degrees, so that the moon 
passes over rather more than its own breadth after entering the 
penumbra, before it reaches the total shadow. This circum- 
stance renders the exact moment of the observed beginning 
or end of a lunar eclipse uncertain. 

208. Effects of the earth's atmosphere. — It is found, by cal- 
culation, that the sun's light which traverses the lowest parts 
of the earth's atmosphere would be so much refracted as to 
meet the axis of the shadow before reaching the moon. Hence, 
the whole disk of the moon is visible, even in a central eclipse, 
and appears of a dull red color. 

. Another effect is the enlargement of the shadow. The light, 
which passes the earth near its surface, and would immediately 
surround the shadow if there were no atmosphere, is, in part, 
obstructed, and in part diffused through the whole breadth of 
the shadow, as just stated. Therefore, the boundary of the 
shadow is enlarged. To make its computed diameter agree 
best with the observed diameter, it is necessary to add ^. 

209. Solar and lunar tables. — In order to determine the 
circumstances of any particular eclipse, tables are needed which 
will give for that time the sun's and moon's hourly motions, 
their parallaxes, and their apparent semi-diameters. Such tables, 
of the most accurate kind, are published in the Nautical Alma- 
nac for each year, and several years in advance. 

8 



114 LUNAR ECLirSE. 

210. The moon? 8 relative orbit. — The center of the earth 9 
shadow moves in the ecliptic at the same rate as the sun, about 
1° per day ; while the moon moves in its orbit about 13° per 
day. To reduce these two motions to one, the relative orbit is 
substituted for the real one, in the following manner. Let 1STG 
*Fig. 58) be an arc of the ecliptic, ~Ng an arc of the moon's 
Drbit, E" the ascending node, A the place of the shadow's cen 
ier, and a that of the moon's center, at the time of opposition 
While A, in one hour, moves to A', suppose a to move to g. 
Then A!g represents the distance and relative direction of the 
centers at the end of an hour after opposition. If gd be drawn 
equal and parallel to A' A, then Ad has the same length and 
direction as A!g. We may, therefore, suppose A, the center oi 
the shadow, to have remained at rest, and a, the moon's center, 
to have moved to d in one hour ; in which case, Yad would be 
the relative orbit, id (= hg) is the moon's hourly motion in lati- 
tude, and ai (= ah — AA') is the difference of hourly motions 
In longitude. 

Fig. 58. 



<* DA' A F^TS*^ 

The inclination of the relative orbit to the ecliptic is found 
by the right-angled triangle dai, in which ai and di being 
known, the angle dai, or its equal, dFD, is computed. 

The change from the true to the relative orbit is greatly ex- 
aggerated in the tigure. If truly represented, ag would be 13 
times as long as A A'. 

211. Times of beginning, middle, and end of a lunar 
eclipse, by projection. — Let ND (Fig. 59) represent an arc of the 
ecliptic, and A the center of the shadow at opposition. With 
any convenient scale of equal parts, lay off from A the minutes 
of hourly motion of the moon from the sun — namely, AB, BC, 
AD, etc., and divide them into as small fractions of an hour as 
is desired. Then draw a circle with the radius Ao, equal tc 



MIDDLE AND OPPOSITION. 



115 



the minutes in the semi-diameter of the shadow. Lay off Aa 
perpendicular to CD, equal to the moon's latitude at opposi- 
tion. Then a is the moon's center at that time. Through a 
draw N/j making N equal to the inclination of the relative 
orbit. Draw Amo perpendicular to N/1 At the middle oi 
the eclipse, the moon's center is at m, because Am bisects the 
chord of the circle. From m draw ml perpendicular to ND. 
The parts of hourly motion between M and A show how long 
before opposition the middle of the eclipse occurs. 

Fig. 59. 




Take a line equal to the sum of the semi-diameters of the 
shadow and the moon, place one end at A, and mark the points 
c and/, with the other end on the moon's path. With a radius 
equal to the semi-diameter of the moon, draw the circles around 
c and/", which will touch the shadow. The eclipse begins 
when the moon's center is at c, and ends when at f. Next 
draw the perpendiculars c-F, /G, and we have on the scale of 
time the interval FA between the beginning and opposition, 
and AG between opposition and the end. 

Finally, if the latitude is so small that the moon falls entirely 
into the shadow, making Ac', A/"', each equal to the difference 
of the two semi-diameters, mark the points c' and/' as before. 
Then the perpendiculars, c'K andy'K, mark the times of the 
beginning and end of the total eclipse. 

212. The middle of the eclipse, how related to the opposi- 
tion. — In the projection just described, N is the ascending 



116 ECLIPSE OF THE SUN. 

node, and the moon passes the node, 1ST, before it reaches the 
opposition, a ; in which case, the middle of the eclipse at m 
precedes the opposition at a. This is true at either node ; the 
middle of the eclipse precedes opposition, if the passage of the 
node precedes it ; but the middle is later than opposition, if the 
passage of the node is later. 

213. Times of beginning, middle, and end of a lunar 
eclipse by computation. — The same results may be obtained 
with greater accuracy by trigonometry. 

As Aa and Am are perpendicular respectively to NT) and 
~Nf, the angle a Am is equal to AN a, the angle of the relative 
orbit. The moon's latitude, Aa, being known, and the angle 
a Am, compute Am ; then by Am and AmM (= aAm) find AM, 
and change it into time by the proportion, hourly motion in 
longitude of moon from shadow : MA : : 1 hour : time of pass- 
ing over MA. Thus, the time of the middle of the eclipse is 
obtained. 

Am and Ac being known, the angle mAc is calculated; 
which subtracted from mA'M (complement of aAm) leave* 
cAE. Hence, in the triangle AcF, Ac and the angle cAF fur- 
nish FA, which, changed to time as before, determines the time 
when the eclipse begins. In the same manner, by the triangle 
Ac'H, the time of the beginning of the total eclipse is found. 
ISTo additional calculation is necessary for the end ; for the in- 
terval between the beginning and middle is equal to that be- 
tween the middle and the end. 

214. Digits eclipsed. — The magnitude of an eclipse, is 
usually expressed in digits, or 12ths of the moon's diameter. 
The distance from n, the inner edge of the moon, to <?, the edge 
of the shadow, is divided into parts, each equal to -^ of nl. 
The number of such parts contained in no expresses the digits 
•eclipsed. If the digits eclipsed equal or exceed 12, the eclipse 
is total. 

215. Eclipse of the sun. — An eclipse of the sun is of a dif 
ferent character from an eclipse of the moon. When the moon 
is eclipsed, it is obscured by the earth's shadow falling on it 



LENGTH OF MOON S SHADOW. 



117 



■ The moon itself is affected. But the sun is said to be eclipsed 
when the moon intervenes between it and the earth, and hides 
it from our view. The sun itself suffers no change, but we are 
placed in circumstances which prevent our seeing it. The 
phenomenon would more properly be called an occultation of 
the sun. 

216. Form and angle of the moon's shadow. — The moon's 
shadow, like the earth's, is a cone, surrounded by a penumbra 
of infinite extent. Let AR (Fig. 60) be the sun, BC the moon, 
and K, the vertex of its conical shadow. The exterior angle 
SDK - DEK + DKR; .-. DKK - SDE — DEK. JS T ow, 
SDE is readily found, being the apparent semi-diameter of the 
sun as seen from the moon. It is larger than as seen from the 
earth, in the inverse ratio of distances, or as 387 : 386, nearly. 
The angle DEK is the sun's horizontal parallax at the moon. 
On account of distance, it is larger than at the earth, nearly in 
the ratio of 387 : 386 ; but on account of the moon's size, it is 
less in the ratio of their diameters, 2161 : 7912. The sun's 
horizontal parallax at the earth, when thus modified, gives the 
angle DEK. Therefore, DKE, the semi-angle of the moon's 
shadow, is found. Its mean value is 16' 1".6, about the same 
as the sun's apparent semi-diameter. 



Fig. 60. 




217. Length of the moon's shadow. — In the triangle DKC, 
right-angled at C, 

sin DKC : rad : : DC : DK, 
the length of the moon's shadow. Its mean length is 231,690 
miles, not quite sufficient to reach to the earth's surface. 

When the moon is nearest to the earth, and the earth at the 



118 SOLAR ECLIPTIC LIMIT. 

same time is furthest from the sun, the shadow is long enough 
to reach about 14,500 miles beyond the earth's center. 

218. Greatest breadth of section on the earth. — In the case- 
just mentioned, if the shadow is directed toward the earth's- 
center, its section at the surface is the greatest possible. To 
find its diameter en, compute the angle eTd, thus: 

eT : TK : : sin eKT : sin TeK, 
and eKT + TeK = eTd. Then, 

360° : eTd : : earth's circumference : ed. 

This, when greatest, is about 85 miles, and therefore the- 
diameter of the section is 170 miles. Within this circle there 
is witnessed a total eclipse of the sun. 

219. The moorfs penumbra, and its greatest section on the 
earth. — The crossing tangents, ACH, RBG, etc., include the 
penumbra. Its semi-angle is BID, which is equal to IKD -f- 
IDR. But IRD is the sun's horizontal parallax at the moon,, 
and IDR is the sun's apparent semi-diameter at the moon. 
Therefore, BID is known. To this add IGD, the moon's appa- 
rent semi-diameter, and the sum equals GDT. Hence, in the tri- 
angle GDT we have GT, TD, and the angle GDT, by which 
GTD is computed. From this, GH, the diameter of the pe- 
numbra on the earth, is obtained, as in the preceding article. 
Its greatest diameter is 4,500 miles. 

220. Solar eclijrtic limit. — The distance of the sun's center 
from the node, when the moon's penumbra at conjunction 
would only touch the earth in passing, is called the solar eclip- 
tic limit. It is obtained by first finding the distance between 
the sun's and moon's centers at the given time. Let S (Fig. 
61) be the sun's center, E the earth's, and M the moon's. It is 
obvious that the limit occurs when the moon's disk just touches 
AB, the extreme solar ray that meets the earth. The angular 
distance between the centers of the sun and moon at that time 
is the angle SEM. But SEM = SEA + AEC + OEM. SEA 
is the sun's semi-diameter = 6. OEM is the moon's semi-diame 
ter = d. The angle AEC (in the triangle EAC) = ECB - OAK. 



MAGNITUDE OF ECLIPSE. 119 

But ECB is the moon's horizontal parallax = P ; and CAE is 
the sun's horizontal parallax = jp. Therefore, the distance be- 
tween the centers, SEM = 6 + d+Y — p ; that is, the sum 
of the semi-diameters of the sun and moon, added to the differ- 
ence of their parallaxes. 

Fig. 61. 




Eepresenting this distance by CM (Fig. 57), CN is computed 
as in Art. 203. At the maximum, it is found to be 18° 36' ; 
and beyond that, an eclipse is impossible. Its minimum value 
is 15° 20' ; and within that, there cannot fail to be an eclipse. 

221. Magnitude of eclipse. — If the eye of the observer were 
at the vertex of the total shadow of the moon, it is plain that 
the moon's disk would exactly cover the sun's. And as the 
moon appears to our unaided vision to be of the same size as 
the sun, this., of itself, shows that the cone of the shadow has a 
length sufficient to reach about to the earth, as proved (Art. 
217). But the moon's semi-diameter is sometimes greater than 
the sun's, and sometimes less. When greater, the eclipse is 
total to all those places which fall within the section of the 
shadow as it crosses the earth. When less, the eclipse is annu- 
lar to places lying sufficiently near the path of the axis of the 
shadow. It is called annular, because a ring of the sun's disk 
is seen about the moon (Fig. 62). An eclipse, whether total oi 
annular, is central at all places where the axis of the shadow 
falls, or to which it points. If only the penumbra passes a 
place, the eclipse there is part ial. The annular eclipse belongs 
to the class of partial eclipses. 

If the total shadow reaches the earth at all, yet its section is 
small, compared with that of the penumbra (Arts. 218 and 



1*20 VELOCITY OF THE SHADOW. 

210). Hence, at a given place, while partial solar eclipses- 
occur frequently, probably one or two every year, a total 
eclipse is extremely rare, perhaps not one in a century. 

Ficr. 62. 




It is possible for an eclipse to be annular to those places 
where it is seen in the morning or evening, and total to those 
in which it is seen near noon ; for on the meridian, the moon 
appears about gV larger than at the horizon (Art. 165), and 
might cover the sun in one case, when it would not in the 
other. If an eclipse thus changes its magnitude from annular' 
to total, and then to annular again, while crossing the earth, it 
results from the fact that the moon's shadow is too long to 
reach the nearest part of the earth's surface, and not long 
enough to reach its center. 

222. Velocity of the thadovj. — The hourly motion of the 
moon from the sun is about 30'. This arc equals 2,080 miles 
of absolute motion of the moon in its orbit. The shadow may 
be considered as having the same velocity as the moon. There- 
fore, the absolute velocity of the moon's shadow on the earth is 
2,08 ( > miles per hour, which is sufficient to carry it across the 
earth's disk in a little less than 4 hours. Relatively to the sur- 
face, the velocity is much less than this. At the equator, the 



SOLAR AND LUNAR ECLIPSES. 12' 

velocity of surface is about 1,040 miles per hour, one-half that 
of the shadow, and both motions are from west to east. Hence, 
at the equator, the shadow passes a place at the rate of about 
1,040 miles per hour, when it falls perpendicularly. When in- 
clined, as at morning and evening, it passes more swiftly, in 
the proportion of radius to the sine of obliquity. The relative 
motion is also greater as the latitude increases, on account ot 
the slower motion of the surface. When an eclipse falls within 
polar circle, the shadow and the observer may possibly move 
in opposite directions, so that the relative motion would be the 
sum, instead of the difference, of the real motions. 

223. Duration of total and a?inular eclipses, — The sun 
and moon differ so little in apparent size, and the velocity of 
the shadow is so great, that the duration of total and annular 
eclipses is necessarily short. It is seen by the preceding article 
that the rotation of the earth generally reduces the relative ve- 
locity ; it therefore increases the duration. The greatest con- 
tinuance of a total eclipse of the sun is about 8 minutes. An 
annular eclipse may continue more than 12 minutes. 

224. Number of solar and lunar eclipses. — If an eclipse of 
the sun occurs in passing each node in a certain year, the lunar 
ecliptic limit is so small, that the moon may escape an eclipse 
at both the previous and the subsequent oppositions. In this 
case, there would be but two eclipses in a year, both solar. 
This is the least number. 

If, however, a lunar eclipse occurs very near a node, the 
solar limit is so large, that there must be one, and there may 
be two solar eclipses at the preceding and following conjunc- 
tions. Thus, there may be as many as six eclipses while the 
sun passes the two nodes. Another one may possibly occur 
before twelve months have elapsed, in consequence of the back- 
ward motion of the nodes. Thus, the greatest number in a 
year is seven, of which five are of the sun, and two of the 
moon. 

225. Relative number of solar and lunar eclipses. — Solar 
eclipses are more numerous than lunar, in the proportion o' 



122 THE SAROS. 

their ecliptic limits — that is, nearly as 3 : 2. But, because one 
is really an eclipse, and the other an occultation, eclipses oi 
the moon at a given place are more frequent than those of 
the sim. An eclipse of the moon is visible to all on the hem- 
isphere nearest to it, without regard to locality. But an 
eclipse of the sun is not seen at a place, unless the moon's 
shadow falls at that place. 

226. Solar and lunar eclipses begin on opposite sides. — As 
the moon moves toward the east much faster than the sun or 
the earth's shadow, we determine on which side of the body 
a solar or a lunar eclipse begins, by simply considering the 
motion of the moon. In a lunar eclipse, the moon overtakes 
the shadow of the earth, and, of course, its eastern limb enters 
the shadow first. Hence, a lunar eclipse always begins on 
the east side of the moon, and ends on the west side. But 
in a solar eclipse, the moon, in its eastward motion, overtakes 
the sun, and conceals its western limb first ; so that a solar 
eclipse begins on the west side of the sun, and ends on the 
east side. 

227. The Saros. — This name is given to the cycle of 18 
years and 10 days, within which there is a return of the 
eclipses of preceding cycles, in the same order, and of nearly 
the same magnitude. The reason for this return of eclipses is, 
that the sun, moon, and node, return to very nearly the same 
relations to each other in the period just named. 

The return of the moon to the sun (a lunation) occurs 223 
times, and the return of the sun to the node (a synodical rev- 
olution of the node) occurs 19 times, in this period of 18 years 
and 10 or 11 days, the two periods differing less than 12 
hours from each other. As the sun, moon, and node, do not 
resume their exact relation to each other, the series of eclipses 
in one cycle will vary a little from those of the preceding; 
and, therefore, after a number of cycles, their magnitude will 
become essentially changed, and at length, one after another, 
they will disappear from the cycle entirely. 

This period was used by the Chaldeans for predicting the 
returns of eclipses, and by them called the Saros. 



PHENOMENA OF A SOLAR ECLIPSE. 123 

228. Phenomena of a total eclipse of the sun. — 

1. The corona. — This is a luminous halo surrounding tho 
moon when the snn is entirely hidden, and sometimes presents 
a radiated appearance, and extends from the moon's edge out- 
ward a distance equal to one-third of its diameter, fading 
gradually to the shade of the sky. It is concentric with the 
sun, rather than with the moon, and is thought to indicate 
an extensive solar atmosphere. 

2. Bailtfs heads. — At the instant when the fine thread oi 
the sun's edge is just appearing or disappearing, it is often 
divided up into a series of separate bright points. Being first 
noticed by Sir Francis Baity, they are known as Baily's beads. 
The appearance is by some attributed to the light of the 
sun's edge coming through between the mountain summits 
of the rough outline of the moon's disk. That they are not 
always seen, may arise from the fact that the limb in con- 
tact may, in some cases, be much less serrated by mountains 
than in others. 

3. Flame-colored protuberances. — Another phenomenon, very 
variable in its aspect, consists of irregular projections, which 
appear here and there around the disk of the sun, after it is 
wholly in occultation. They are sometimes broad, and of small 
elevation ; at others, they extend out nearly a tenth of the di- 
ameter of the sun — that is, to the height of 80,000 miles, and 
are often bent at a considerable angle. Occasionally, they are 
entirely detached from the disk. These flame-colored or rose- 
colored prominences, when first discovered, were not supposed 
to be sufficiently luminous to be seen except when the sun was 
wholly covered by the moon. But improved instruments have 
more recently rendered them visible at other times. They are 
found to consist mainly of red-hot hydrogen : thrown violently 
upward from the fiery surface of the sun. 

A total eclipse of the sun is one of the most Sublime and im- 
pressive phenomena of nature. The darkness is such, that the 
larger planets and stars appear ; and yet it is surprisingly sud- 
den in coming and going ; for within a few seconds before and 
after the total darkness, the light is equal to that of hundreds 
of full moons. A chill is felt like that of night. It is not 
strange that people of barbarous countries are filled with con- 



124 CALCULATION JF ECLIPSES. 

sternation and fear by the occurrence of a total eclipse of the 
sun. Appendix D. 

229. Eclipses at the moon. — When we witness a solar 
eclipse, a spectator at the moon would notice only a small, 
dimly-defined circular shadow passing over the earth's disk. 
It would be a partial eclipse of the earth. 

But when we see a total lunar eclipse, the phenomenon at 
the moon would be one of great interest, and of very strange 
appearance. A dim red light from all parts of the sun's disk 
is spread over the moon, being refracted thither by the earth's 
atmosphere (Art. 208). Hence, a spectator there would see the 
sun expanded out into a thin dull-red ring, surrounding the 
earth, and, therefore, having nearly four times the usual diam- 
eter of the sun's disk. 

230. True form of shadows. — It is impossible, in ordinary 
diagrams, to present the shadows of the earth and moon in 
their true proportions. The distance of the sun is so very 
great, compared with its diameter, that the shadows are ex- 
ceedingly slender, having a length about 11 times the diame- 
ter of the base. Fig. 63 is intended to exhibit them in their 
true forms. A is the earth, and B the moon, having just 
emerged from an eclipse. Only one-half of the whole length 
of the shadow of each is presented. Again, on another scale, 
C is the moon, and D the earth, on which its shadow is falling 
in a solar eclipse. 

Fig. 63. 

A B l 

( — ■— ^— I 



231. Calculation of eclipses.— -Particular instructions are 
given in various works on practical astronomy for calculating 
all the circumstances of a solar or a lunar eclipse. Such in- 
structions, with examples for illustration, may be found in 
Loomis's Practical Astronomy and Coffin's Solar and Lunar 
Eclipses. 



LONGITUDE BY THE CHRONOMETER. 125 



CHAPTEE XIII. 

METHODS OF DETERMINING TERRESTRIAL LONGITUDE. 

232. Local time. — Time is reckoned at every place from 
the moment when the sun crosses the meridian at either the 
upper or the lower culmination. This is called local time ; for 
at the same absolute instant, the time thus reckoned at any 
place differs from that on every other meridian. 

233. Connection between longitude and local time. — The 
earth turns uniformly on its axis toward the east through 15° 
every hour. Therefore, a place lying eastward of another will 
have the sun earlier on its meridian, and consequently, in 
respect to the hour of the day, will be in advance of the other 
at the rate of one hour for every 15°. Thus, to a place 15* 
east of Greenwich observatory, it is 1 o'clock p. m. when it is 
noon at Greenwich ; and to a place 15° west of that meridian, 
it is 11 o'clock a. m. at the same instant. Hence, the differ- 
ence of local time at any two places indicates their difference 
of longitude. 

234. Longitude by the chronometer. — If a person leaves 
London with a chronometer accurately adjusted to Greenwich 
time, and travels eastward till he finds his own time slower 
than the local time of the place by lh. 30m., then he know? 
the place to be 22° 30' E. longitude. For 15° x 1£ == 22^°. 
On the contrary, if he travels westward, and at length finds 
his time-piece at 6h. 44m., when the local time is 4h. 32m. — in 
other words, that his Greenwich time is 2h. 12m. too fast — then 
the longitude of the place is 33° "W. In the same manner, 
the longitude of any two places may be compared with each 
other. 

For the use of navigators, chronometers are made which run 
with very great accuracy, and may be relied on during long 



126 LONGITUDE BY A SOLAE ECLIPSE. 

voyages. There is always a probability, however, that a chro- 
nometer may change its rate somewhat, when it comes to be 
transported from place to place. It is therefore safer on long 
voyages to nse several chronometers, and employ the mean of 
all their indications. 

235. Longitude by a lunar eclipse. — In one respect, a lunar 
eclipse is very favorable for the comparison of longitudes. It 
is a distant phenomenon, seen at the same absolute instant by 
all. Hence, any difference of time in the observations at dif- 
ferent places is entirely due to difference of longitude. 

But in another respect, it is quite unfitted for the purpose. 
On account of the penumbra, there is no definite edge to the 
shadow which passes over the moon's disk, and consequently 
there is great uncertainty as to the time of beginning or end of 
the eclipse. This method is but little depended on for accurate 
results. 

236. Longitude by a solar eclipse. — In both the above par- 
ticulars, a solar eclipse differs from a lunar. It is not an evenl 
at a distance, seen at once by all, but on the earth's surface, 
happening to one place at one instant, and to another place at 
another. The time of beginning or end of a solar eclipse de- 
pends on the position of the observer. 

On the other hand, the phenomenon is very definite, and 
the moments of immersion and emersion of the sun's limb can 
be quite accurately fixed by observation. 

To compare longitudes by a solar eclipse, the observations 
made on the beginning and end at a given place are used as 
means of calculating the time of conjunction — that is, the time 
when the sun and moon are in the same secondary of the 
ecliptic. But that event occurs at a certain absolute instant. 
This computation being made for each place, the time of con- 
junction ought to be exactly the same, so that the difference in 
the results is wholly due to a difference in the longitude of the 
places. This method of obtaining the longitude of a place is 
accurate, but laborious. 

Occultations of stars by the moon are much more frequent 
than the occultation of the sun ; and these are phenomena of 



LONGITUDE BY THE TELEGRAPH. 127 

the same general character, and may be used in the same way 
for finding the longitude of a place. 

237. Longitude by eclipses of Jupiter's satellites. — The sat- 
ellites of Jupiter fall into the shadow of that planet, as the 
moon does into the shadow of the earth. Every such eclipse 
occurs at a certain time ; and all who see it, see it at the same 
instant. Hence, these eclipses are favorable for determining 
longitudes. Moreover, they are occurring every day, while 
eclipses of the sun and moon are rare. 

But, on account of the penumbra of the planet, and the con- 
siderable diameter of the satellites, they disappear and reappear 
gradually. There is difficulty, therefore, in observing accu- 
rately the beginning and end of these eclipses. In order to 
obtain the best results, the telescopes used by different ob- 
servers ought to be alike in aperture and power. 

238. Longitude by the lunar method. — This is a method 
particularly useful to navigators, because the observations are 
made by the sextant. It consists in measuring the angular 
distance between the moon and some conspicuous heavenly 
body, as the sun, or a large planet or star, and then correct- 
ing the observation for parallax and refraction, so as to have 
the true distance between the bodies, as seen from the center 
of the earth. The observer must also note the local time 
when this measurement is made. 

Having with him the Nautical Almanac, in which the dis- 
tances, as seen from the earth's center, are predicted for every 
•lay and hour of Greenwich time, he looks for the Greenwich 
time at which the distance agrees with the distance as he has 
obtained it. The absolute time is the same : hence, the dif- 
ference of time shows his longitude from Greenwich. 

The bodies, whose angular distances from the moon the 
Nautical Almanac gives for every three hours, with propor- 
tional numbers for interpolation, are the sun, Yenus, Mars, 
Jupiter, Saturn, and nine bright fixed stars. 

239. Longitude ~by the telegraph. — Since the invention oi 
the magnetic telegraph, it has been employed to determine the 



12S VELOCITY OF ELECTRIC CURRENT. 

differences of longitude between fixed stations on land with a 
precision which was before altogether unattainable. Suppose 
two stations to be connected by the telegraphic line, and that 
there is at each a clock keeping the local time. The observ- 
ers agree beforehand at what time, by his own clock, the one 
at the most easterly station shall commence giving signals ; 
and also at what time the other shall commence giving another 
series according to his clock. The interval between successive 
signals is also previously determined. When the moment ar- 
rives, the first observer strikes the telegraphic key at the ex- 
act beat of the clock, and the second observer records the 
time of the signal as shown by his own clock ; and thus they 
continue to do till the full series is recorded. The second ob- 
server then commences sending signals, which are in like 
manner recorded by the first. The velocity of the electric 
current is so great, that the absolute time of making a signal 
at one station, and of perceiving it at the other, may be con- 
sidered identical ; so that the difference which is indicated 
by the two clocks in each case is wholly due to difference 
of longitude. Still greater precision is attained by causing 
the signal key at each station to record its own movement on 
the line of second-marks made by the clock at the other sta- 
tion (Art. 46). 

240. Velocity of the electric current. — The method just de- 
scribed is susceptible of such accuracy, that it has led to the 
discovery of the velocity of the current. For, if the moment 
of its arrival at the distant station is not identical with that 
of the signal given, it will indicate a difference of longitude 
less than the true difference when sent westward, but greater 
than the true difference when sent eastward. By this discrep- 
ancy, if it is appreciable, the velocity of the current becomes 
known. It is found to be about 16,000 miles per second. 

241. Change of days in circumnavigating the earth. — While 
a person travels westward, he lengthens his days by one hour 
for every 15°, or 4 minutes for every degree, since he moves 
along with the apparent diurnal motion of the sun. In travel- 
ing eastward, on the contrary, he shortens the days at the same 






DAYS IN THE PACIFIC OCEAN. 129 

rate, by moving in opposition to the sun's daily progress. Ii 
we suppose him to go westward entirely round the earth to the 
same meridian again, whether he takes a longer or a shorter 
time for the journey, he will lengthen the individual days suf- 
ficiently to make the whole number just one day less than if he 
had remained where he was. The 5th of a month is fo him the 
4th ; and Tuesday, according to his reckoning, is Monday. 
The reason is obvious ; for during his journey, the earth has 
made a certain number of diurnal revolutions from west to 
east ; but he, by going round from east to west, has, in respect 
to himself, diminished that number by one. 

All this is exactly reversed when one goes round the globe 
from west to east. He gains just a day by making all the days 
of his travel a little shorter. It is plain that he makes one 
more diurnal revolution from west to east than the earth 
does. 

Of course, if these two individuals meet at their place of 
starting, they differ from each other just two days in their 
reckoning. 

242. Ambiguity as to days among the islands of the Pacific 
Ocean. — If an island in the Pacific were settled by navigators, 
who had gone westward around Cape Horn, and also by others, 
who had sailed eastward around the Cape of Good Hope, the 
reckoning of these two parties would differ by one day. To 
the former, a day will be the first of a month when it is the 2d 
to the latter. It is, in fact, true that there are islands lying 
contiguous to each other which have this difference of reckon- 
ing. ^ 

If inhabited land extended entirely round the earth, it would 
be necessary to Hx arbitrarily on some meridian on which the 
change of day should be made. For it is impossible that the 
reckoning of days should go on unbroken around the earth. 
The arbitrary meridian would separate between places which 
differ a day from each other ; so that, on the west side of it, 
the time is one day later, both in the month and the week, 
than on the east side. 



180 



WATER ACTED OX BY THE MOON. 



CHAPTEE XIV. 



THE TIDES. 



243. Definitions. — The tides are the dailj rising and fall- 
ing of the waters of the ocean. When the water, in this datt, 
oscillation, has reached its highest point, it is called high- 
water • at its lowest point, it is called low-water. While the 
water is rising, it is called flood; and while falling, ebb. 

A lunar day is the time between two successive culmina- 
tions of the moon. Its length is about 24h. 52m., being nearly 
an hour longer than a solar day on account of the rapid east- 
ward motion of the moon. The tides make their revolutions 
within the lunar day. 

Twice in a lunation high-water is at a maximum, and twice 
it is at a minimum; the former are called spring tides ; the 
latter, neap tides. The spring tides occur near the time o+ 
syzygies, the neap tides near the time of quadratures. 

244. Opposite tides. — There are two tide-waves on opposite 
sides of the globe, moving around it from east to west, and ar- 
riving at any place at intervals, whose mean value is 12h. 
26m., or half a lunar day. Since the mean diurnal motion of 
each of the two opposite tides is the same as that of the moon, 
the action of the moon must be regarded as the principal cause 
of the tides. 



245. Form of the water acted on 
by the moon. — If the earth were cov- 
ered with water, and no force were 
exerted except gravitation toward the 
earth itself, its form would be exactly 
spherical, as represented in Fig. 64. 
But if a distant body, as the moon, 
should also attract it, the sphere would 
be changed into a prolate spheroid — 
that is, into a form produced by re- 
volving an ellipse about ? ts major axis. 




Let the moon be id 



JOINT ACTION OF THE SUN AND MOON. 131 

ihe direction of CE produced, and suppose the center of gravity 
of the nearer half of the water, DEF, to be at A, and that 01 
the remote half at B, while the center of the earth, as a whole, 
is at C. Since A is more attracted than C, and C more than 
B, the form of equilibrium must be disturbed, and some of the 
water will flow toward E, and other parts toward G, till the 
particles are in equilibrio between their unequal tendencies to 
the moon, and their gravity on the inclined surface of the 
spheroid. E and G are the highest points of the spheroid, and 
all points on the circle DF (perpendicular to EG) are the 
lowest. Every section through EG is an ellipse, whose major 
axis is EG, and whose minor axis is equal to DF. The ellip- 
ticity of the section will obviously depend not only on the 
strength of the moon s attraction, but also on the difference be- 
tween the attractions on the nearer and remoter parts. 

In the case of the earth and moon, it is computed that the 
major axis would exceed the minor by 5 feet — that is, the tides 
would be only 2\ feet high, and on opposite sides of the 
earth, one directed toward the moon, the other from it. The 
tide on the side nearest the moon is sometimes called the direct 
tide ; the one on the remote side, the opposite tide. 

246. Tides oy the sun. — The same kind of effect is pro- 
duced by the sun as by the moon. But the distance of the sun 
is so great, that though it attracts the eartb more than the 
moon does, yet the difference of its attractions on the several 
parts is less. The power of the moon to raise a tide is to that 
of the sun about as 5 to 2. 

247. Joint action of the sun and moon. — At the time of 
conjunction, the moon and sun attract in the same direction, 
and therefore the tides are equal to the sum of the lunar and 
solar tides. The same is true at opposition, because each body 
produces two tides at once ; and the direct lunar tide coincides 
with the opposite solar tide, and vice versa. These are the 
spring tides which occur at the syzygies. 

At quadratures, each body raises a tide at the expense of 
that raised by the other. For if the moon is in the direction 
of EG produced (Fig. 64), it causes high-water at E and G, 



132 



DIURNAL INEQUALITY. 



and low-water at D and F. And if the sun is in the direction 
of DF produced, it causes high-water at D and F, and low- 
water at E and Gr, As the lunar tides are the highest, E and 
G are the neap tides, made less by this action of the sun, than 
if the moon had acted alone. 

248. Effect of the inertia of water. — If the moon and earth 
were at rest, the tides would be directed exactly to and from 
the moon. But while the waters are flowing toward these 
points, the moon, by the diurnal motion, passes westward, and 
causes them to change the places at which they tend to accu- 
mulate. Thus, even if the wave were unchecked by the shores 
of continents and islands, the summit would be two or three 
hours behind the moon in passing a given meridian. 

249. Diurnal inequality. — At a given place, the two tides 
which follow the culmination of the moon will vary in height, 
according to the relation between the latitude of the place and 
the moon's declination. If the moon, M. (Fig. 65), is on the 
equator, it is clear that the tides on the equator, EQ, are great- 
est, and that in other places they are less, as the latitude is 
greater. But the two successive tides at any place are equal ; 
for, by the rotation on NS, the tide at B in 12 J hours will 
come round to A, and be equal to the tide now there. The 
same is true of the tides C and D, or F and Gr. Hence, when 
the moon has no declination, there is no diurnal inequality. 




But suppose the moon has a northern declination, as in Fig 
66. Then the highest points of the tide are at A in north lat 



COTIDAL LINES. 



133 



itude, and D in south. At A, where the direct tide is large, 
the opposite tide now at B will arrive in 12£ hours, and will 
be small. But at C, this is reversed ; the direct tide is small, 
and the opposite one (now at D, and arriving at C 12J hours 
later), is large. Therefore, when the decimation and the lat- 
itude are both north, or both south, the direct tide — that is, the 
tide which first succeeds the upper culmination of the moon — 
is larger than the opposite tide ; but if one is north, and the 
other south, the direct tide is smaller than the opposite tide. 
This difference in the height of the two successive tides is 
called the diurnal inequality. 



250. Change of direction and velocity caused by coasts. — 
The tide-wave, which would move regularly westward around 
the earth, if it were wholly covered by deep water, is exceed- 
ingly broken up and changed, both in direction and velocity, 
by coasts and shoals. Its general direction is westward ; but 
as it can pass the continents only at their southern extremities, 
it bears to the northwest, and then to the north, in the Atlan- 
tic and Pacific oceans ; and when it enters seas or channels, i1 
usually bends its course in the direction of their length. 



Fig. 67. 



251. Cotidal lines. — These are lines drawn on a chart ol 
the oceans, showing the posi- 
tion of the summit of the tide- 
wave for each hour of a day. 
Such a system of lines expresses 
to the eye the direction and ve- 
locity of the tide at all places. 
Thus, on the open ocean, the 
figures 1, 2, 3, 4 (Fig. 67) show 
the situation of one and the 
same tide-wave at those hours, 
respectively. And in the chan- 
nel which extends northward, 
the wave, having separated from 
the ocean tide, advances north- 
ward, and occupies the places 
marked at the hours indicated. The wave advances most rap 




134 ESTABLISHMENT OF A PORT. 

idly in the deepest water. Hence, the front is generally convex 
as in Fig. 67, since it moves fastest in the central part, where 
the water is deepest. For this reason, also, the tide may occu- 
py as long a time in running through a long channel of shal- 
low water as in advancing half round the earth. The greatest 
velocity of the tide in the deep, open ocean, is near 1,000 miles 
per hour. Some channels are affected by tides entering at both 
extremities. For example, the German Ocean and English 
Channel receive the Atlantic tide both at the north and at the 
south end. As a consequence, the tide system is doubled, 
causing, at some points, four tides per day. 

252. Modification in the height of the tide caused by 
coasts. — The relation of coast lines to each other also very 
much affects the height of the tide at particular places. When 
the tide directly enters a broad-mouthed bay, it grows higher 
as the bay contracts in breadth ; and at the head of the bay, 
there is usually found the greatest height of all. One of the 
most remarkable examples is the Bay of Fundy. The western- 
extremity of the Atlantic tide-wave, after entering this bay, is 
gradually contracted by the shores as it advances, till, at the 
head of the bay, it sometimes rises to 70 feet. 

The height of the tide on the coast is generally greater than in 
the open ocean, owing to the effect of shoal water. The most 
advanced part of the wave moves slower than the hinder por- 
tion ; so that the cross-section of the ridge becomes shorter y 
and therefore higher, as the depth of water diminishes. 

The mean height of the spring tides at any place is called 
the unit of altitude for that place. 

253. Establishment of a port. — This phrase signifies the 
mean interval between the culmination of the moon and the 
arrival of the tide at a given place. At every meridian, the 
tide arrives later than the body which causes it ; but the delay 
varies exceedingly at different localities, on account of shoal 
water, direction and length of channel, etc. Even at the same 
place, the delay during a lunation varies according as the 
small solar tide precedes or follows the large lunar one ; for 
the summit lies between them. It is the mean interval at 



TIDES MODIFIED BY DISTANCE. 135 

a given port, which is called the establishment of that 
port. 

254. Tides of lakes and inland seas. — In general, the tides 
of lakes and inland seas are scarcely perceptible. The reason 
is, their extent is so small, that all parts are to be considered as 
almost equidistant from the moon. There is little opportunity 
for water to be attracted from the more distant to the nearer 
part. The largest North American lakes have tides but an 
inch or two in height. In the Mediterranean, however, which 
derives no tide from the ocean, the tide-wave reaches 1J or 2 
feet. 

25 5. Tides modified by the sun's and moon's change of dis- 
tance. — The difference of the moon's attraction on the several 
parts of the earth is greatest when the moon is nearest, and 
least when it is most distant. The same is true of the sun. 
Hence, the tides of each month have a periodical increase and 
decrease as the moon passes through its perigee and apogee. 
They have a like, though much smaller, change each year, at 
the perihelion and aphelion of the earth's orbit. By the revo- 
lution of the apsides of the moon's orbit, these maxima and 
minima will alternately coincide once in 9 years. Combining 
these changes with those at syzygy and quadrature, the height 
of the greatest possible spring tide, to that of the least possible 
neap tide, is as 10 to 3. 



L36 CLASSIFICATION OF THE D LANETS 



CHAPTEK XV. 

THE PLANETS. — TABULAR STATEMENTS. — MERCURY. — VENU& 

MARS. 

256. Names and classification of the planets. —The planets 

are solid spherical bodies revolving about the sun in orbits 
which are nearly circular. The name " planet'' ' signifies a 
wanderer, and was given to these bodies because they con- 
tinually change their places among the fixed stars, generally 
moving from west to east, but sometimes from east to west. 
These apparently irregular motions are fully explained by our 
own annual motion, the earth on which we live being one of 
the planets. 

The planets are naturally arranged in three classes. 

1. Four small planets near the sun, of which the earth is the 
largest — namely, Mercury, Venus, Earth, Mars. 

2. The planetoids, an indefinite number of bodies, too smal" 
to be measured with certainty, and occupying a ring outside* 
of the first class. They are also called asteroids, and minor 
planets. 

3. Four large planets, moving outside of the ring of plan- 
etoids, widely separated from each other, and at vast distances- 
from the sun. These are Jupiter, Saturn, Uranus^ Neptune. 

Two planets of the first class, Mercury and Yen us, revolve 
in orbits within the earth's orbit. These are called inferior 
planets, being lower down in the solar system than the earth 
is. All the others, including the planetoids, are called superior 
planets ; because, in relation to the sun, the great center of at- 
traction, they are higher than the earth, and revolve in orbits 
exterior to the earth's orbit. Appendix F. 

257. Satellites. There is another class of spherical bodies, 
holding a subordinate place in the solar system, since they re- 
volve around the planets as centers. These are called satel- 
lites The moon, already described in Chapter X,, is a satellite 



PERIODIC TIMES OF PLANETS. 



137 



of the earth. They are distributed as follows : the Earth has 1 ; 
Mars, 2; Jupiter, 4; Saturn, 8 ; Uranus, 4 ; Neptune, 1. Mercury 
and Venus have no satellites. 

The satellites are also called secondary planets : and the 
planets, in distinction from them, primary planets. 

258. Distances of the planets from the sun. — The follow- 
ing table presents the mean distances of the planets from the 
sun in millions of miles, and also their relative distances, the 
earth's being called 1. 





Mean Distances. 


Relative 
Distances. 


Mercury 


36,000,000 

67,000,000 

92,000,000 

141,000,000 

250,000,000 

481,000,000 

881,000,000 

1772,000,000 

2775,000,000 


0.39 
0.72 
1.00 
1.52 
2.67 
5.20 
9.54 
19.18 
30.05 


Venus 


Earth 


Mars 


Planetoids 

Jupiter 


Saturn 


Uranus 


Neptune 





It appears by this table, that the remotest planet is 77 times 
as far from the sun as the nearest. Hence it is that orreries, 
unless of inconvenient size, always fail of truly representing 
the planetary distances. The same is generally true of dia- 
grams. 



259. Periodic times of the planets, — The following table 
contains the length of the sidereal revolutions in months and 
years, which is the most convenient form for the memory ; 
their length in days and decimals, for calculations ; their mean 
daily motion ; and the time of their diurnal rotations, so far as 
known, in hours and decimals. 



138 



MAGNITUDES. 



II. 



Mercury. . 
Yenus .... 

Earth 

Mars 

Planetoids 
Jupiter . . . 
Saturn . . . 
Uranus . . . 
Neptune . 



Sidereal Revolu- 
tion. 



3 months, 
7J " 

1 year. 

2 " 

4.1 a 

12 " 

29 " 

84 " 

165 " 



Sidereal Revolu- 
tion in Days. 



87.969 
224.701 
365.256 
686.980 

4332.554 
10759.104 

30686.246 
60228.072 



Mean Daily 


Diurnal Ro- 




Motion. 


tation. 


4° 


5' 32".5 


24.09 h. 


1° 


36' 7".7 


23.35 " 


0° 


59' 8".3 


23.93 " 


0° 


31' 26".5 


24.66 " 


0° 


4' 59".l 


9.92 " 


0° 


2' 0".5 


10.24 " 


0° 


0' 42".2 




0° 


0' 21".5 





It will be found, by comparing the squares of any two periods 
in Table II, and the cubes of the corresponding distances in 
Table I, that their ratios are nearly the same ; and this should be 
true according to Kepler's third law (Art. 119). Thus, for Nep- 
tune and the earth, 30 3 : I 3 = 27,000 ; and 165 2 : l 2 = 27,225. 
So also, while Neptune is 77 times as far from the sun as Mer 
cury is, its period of revolution is 685 times as long. For 
77 3 : l 3 : : 685 2 : l 2 , nearly. 

Since the periods increase more rapidly than the radii of the 
orbits, the velocities of the planets must become less, the fur 
ther they are from the sun. The distance described by Mer- 
cury in a day is nearly nine times that which Neptune passe9 
over in the same time. 

Ill 





Diameters. 


Apparent 
Diameters. 


Volumes. 


Sun 


860,000 

2,992 

7,660 

7,918 

4,211 

86,000 

70,500 

31,700 

34,500 


32' 4" 

0' 7" 

17" 

9" 

37" 

16" 

4" 

3" 


1,295,000.000 

0.054 

0.880 

1.000 

0.248 

1,350.000 

689.000 

75.000 

102.000 


Mercury 

Venus 


Earth 


Mars 


Jupiter 


Saturn 


Uranus 


Neptune 



MAGNITUDES. 



139 



260. Magnitudes of the planets. — Table III gives the di- 
ameters of the sim and planets in miles, their mean apparent 
diameters, and their volumes compared with the earth. 

In comparing the n ambers of this table, it is noticeable that 
in general the planets diminish in size in each direction from 
the planetoids. If we suppose Mars to be placed between 
Mercury and Yenus, and Uranus and Neptune to change 
places with each other, this would be strictly true. 

Observe also that the diameters of the large planets beyond 
the planetoids are from eight to eleven times as large respec- 
tively as those of the small ones within that group. Thus, 



Diam. of Jupiter 
" Saturn 
" Neptune 
" Uranus 
and the sum 



11 
9 



11 
10 



nearly. 



that of the earth 
" Yenus 
" Mars 
" Mercury 
the sum 

Another remarkable fact appears on comparing the diameters 
in Table III, and the times of diurnal rotation in Table II. 
The four small planets all rotate in periods of about 24 hours. 
But the large planets, so far as known, revolve in about 10 
hours. Hence, the equatorial velocity of rotation is far greater 
on the large than on the small planets. That on Jupiter, for 
example, is 27 times as great as that on the earth. 

The dimensions of the planetoids are not given in the table, 
being too small for measurement. One or two of the largest 
are thought to be from 100 to 200 miles in diameter. 

IY. 





Masses. 


Density. 


Specific 
Gravity. 


Sun 


326,800.000 

0.065 

0.769 

1.000 

0.111 

311.953 

93.329 

14,460 

16.862 


0.25 
1.21 

0.85 
1.00 
0.73 
0.24 
0.13 
0.22 
0.20 


14 

6.8 
5.2 
5.5 
4.2 
1.3 
0.8 
1.3 
0.9 


Mercury 

Yenus 


Earth 


Mars 


Jupiter 


Saturn 


Uranus 


Neptune 



140 DIAMETERS OF PLANETS. 

261. Masses and densities of the planets. — Table IV ex- 
hibits the masses and densities of the sun and planets, the 
earth being called 1 ; also their specific gravities. 

It appears from table TV, that the small planets are mnch 
more dense than the large planets and the sun. 

262. The sun and planets compared. — By Table III, we see 
that the sun has 10 times the diameter, and 1,000 times the 
volume of Jupiter, the largest planet in the system. Table IV 
shows that the mass of the sun is also more than 1 ,000 times as 
great as that of Jupiter, and 700 times greater than the united 
masses of all the planets. Its attraction mainly controls the 
movements of all the planets, satellites, and comets. Hence, 
these bodies describe their various paths about it, scarcely dis- 
turbing it from a state of rest. For this reason, this system of 
bodies is called the solar system. 

263. Diameters of planets, and their distances from the 
sun. — One of the most remarkable facts relating to the planets 
is brought to view in comparing the distances in Table I with 
the diameters in Table III. While the diameters of the planets 
are only a few thousands of miles, their distances from the sun 
are many millions. The diameter of Neptune's orbit is more 
than 20,000 times the diameters of all the planets added to- 
gether. To attempt to represent both the distances and mag- 
nitudes of the planets in their proportions, by an orrery or 
diagram, is out of the question. 

264. Directions of the planetary motions.* — It has been 

* It is desirable that the student should be able to recognize the planets, and 
become familiar with their motions. Some aid can be had by the use of the 
common almanacs. The "American Ephemeris and Nautical Almanac," pub- 
lished annually at Washington, can be purchased for one dollar by applying to 
the B i: eau of Navigation. This gives the exact places of the sun, moon, and 
planets for each day of the, year. 

The orrery, heliotellus, and lunatellus are instruments that explain their 
motions. The planisphere is an inexpensive instrument that shows the places of 
several hundred of the more conspicuous fixed stars. It can be readily adjusted 
for any hour of the night. The astral lantern is a similar device, of larger size, 
for exhibiting maps of the stars on the illuminated sides of a cubical box. 
Appendix M. 



APPARENT MOTIONS OF MERCURY. 



141 



eUteJ in preceding chapters that all the motions of the sun, 
earth, and moon are from west to east. The same thing ia 
true, in general, of all the planets and satellites ; and in nearly 
every case the inclination to the ecliptic is very small. The 
only exceptions are found in the satellites of Uranus and Nep- 
tune, whose planes of revolution are nearly perpendicular to 
the ecliptic, and the motion in them from east to west. All 
the planetoids yet discovered revolve from west to east, though 
the orbit of one of them has an inclination as large as 34°. 

Since the motions in the solar system are so generally from 
west to east, this is regarded as direct motion ; and any mo 
tions, real or apparent, which are from east to west, are called 
retrograde. 

MERCURY. 

265. Tabular statements. — Mean distance from the san. 
35,761,000 miles; periodic time, 3 months; diameter, 2,993 
miles; diurnal rotation, 24.09 hours; specific gravity, 6-8. 

Fig. 68. 




266. Apparent motions.—Mercury is an inferior p!anet, 
whose orbit is far within the earth's ; for it is seen alternately 
east and west of the sun, and never more than 29° from it. 
Let E (Fig. 68) be the earth, supposed, for the present, to be at 



142 



APPARENT MOTIONS OF MERCURY. 



rest ; the circle ABD, the orbit of Mercury ; S, the sun ; and 
B'A', the sky, on which the bodies are seen projected. When 
Mercury is at B, it is seen at B' ; as it passes through D to A, 
it appears to advance to A' ; as it is now coming toward the 
earth, it seems to be stationary at A' ; then from A through C 
to B, it appears to retrograde from A' to B', where it is again 
stationary, as it moves away from us. Since the sun appears at 
S'j the planet passes by it, both when advancing and when 
retrograding. 

When the planet is at D and C, it is in conjunction with the 
sun ; at C, between the earth and sun, it is said to be in the 
inferior conjunction ; at D, in superior conjunction. B and A 
are called the points of greatest elongation. At superior con- 
junction, the motion of Mercury appears to be forward ; at the 
inferior conjunction, backward ; and if the earth were at rest, 
as we are now supposing, the planet would appear stationary 
at the points of greatest elongation. 



Fig. 69. 




267. The motions of Mercury as modified by the earth's 
motion. — To simplify the case, it was supposed, in the preced- 
ing article, that the earth is at rest. But the earth moves in 



SYNODICAL PERIOD OF MERCURY. 143 

nearly the same direction as Mercury, making about one rev- 
olution while Mercury makes four (Table II). The effect is to 
lengthen the arc of apparent advance, and shorten that of re- 
trogradation. Thus, let the earth be at A (Fig. 69), when 
Mercury is at F ; then it will appear in the sky at L. While 
the earth is advancing to B, Mercury passes the inferior con- 
junction, and arrives at G, and appears at M, having moved 
apparently backward from L to M. As the earth moves to C, 
Mercury describes GKH, and is at superior conjunction ~N. 
Again, while the earth moves to D, Mercury passes round to 
G, still advancing in the sky to O. But while the earth de- 
scribes DE, Mercury again passes the inferior conjunction from 
G to K, and apparently retrogrades from O to P ; after which, 
it begins once more to advance. Thus, by the earth's motion, 
the planet is made to retrograde through a shorter arc, and to 
advance through a longer one, than if the earth were at rest. 

268. Stationary points. — If the earth were at rest, as sup- 
posed in Fig. 68, the points where the planet would appear 
stationary, in relation to the stars, would be A and B, at which 
tangents drawn from the earth would meet the orbit. But the 
earth's motion removes the apparently stationary points a little 
way toward the inferior conjunction. For, in order to appear 
stationary, the advance which the earth's motion causes, must 
be just neutralized by the retrogradation of Mercury. This 
planet appears stationary, when its elongation from the sun 
is 15° or 20°, according as it is nearer the perihelion or the 
aphelion. 

269. The synodical period of Mercury. — This is the time 
in which it goes from a conjunction to the next conjunction of 
the same kind — that is, describes one revolution relatively to 
the earth instead of a star. 

The sidereal period having been obtained by observing the 
planet's return to its node, the synodical period can be com- 
puted from it by using the relative motions of Mercury and the 
earth, just as we find the time in which the minute-hand of a 
watch wiL overtake the hour-hand. The synodical period oi 
Mercury can also be found independently, by means of transits 



144 TRANSITS OF MERCURY. 

across the sun's disk. The synodical period of Mercury is llfi 
days, which is nearly a month longer than its sidereal period. 

270. Form and position of Mercury's orbit. — The orbit 01 
Mercury is more eccentric, and more inclined to the ecliptic 
than that of any other of the eight planets. While the eccen- 
tricity of the earth's orbit is only J ff , that of Mercury is nearly 
\. Yet this renders the minor only ^ shorter than the major 
axis ; so that the form of the most eccentric of *Jie planetary 
orbits, if correctly drawn, would appear to the eye to be a 
circle. 

The inclination of Mercury's orbit to the plane of the eclip- 



271. Phases of Mercury. — At the inferior conjunction, C 
(Fig. 6S). the unilluminated side of Mercury is turned toward 
the earth, so that, like the n^w moon, it is invisible. At the 
superior conjunction, D, its illuminated side is toward us, anc 
it is full. At A or B, where the ray AS, and our line of vis- 
ion, AE, are at right angles, the phase is a semicircle. On the 
arc AOB occur the crescent phases : on BDA, the gibbous 
phases. 

272. Point of greatest brightness. — Mercury is not bright- 
est when full, because it is then too far distant. It is not 
brightest when nearest, because its dark side is toward us. 
Nor is it brightest at the place of greatest elongation ; but 
beyond it, toward the superior conjunction, when about 22° 
from the sun. Its apparent diameter, when nearest the earth, 
and when most distant from it, is as 2£ to 1. 

273. Transits of Mercury. — As Mercury, at the inferior 
conjunction, passes nearly between the earth and sun, it may 
possibly come exactly in a line with them, and thus be seen as 
a black round spot going across the sun's disk. This phenom- 
enon is called a transit of Mercury. If the plane of its orbit 
were coincident with that of the ecliptic, a transit would 
obviously occur at every inferior conjunction. Since the angle 
between the two planes is 7°, the planet can not.be seen on 



APPAEENT MOTIONS OF VENUS. 145 

the disk, unless near the node, for its perpendicular distance 
from the ecliptic must be less than the sun's apparent semi- 
diameter — that is, less than 16'. By a simple calculation, like 
that in Art. 203, it is found that the limit of transit for Mer- 
cury is 2° 10'. 

274. Node months for Mercury. — The nodes of Mercury's 
orbit lie in that part of the heavens which the sun passes 
through in May and November. Therefore, a transit of that 
planet can occur only in those months. More transits happen 
in November than in May, because the planet is nearer peri- 
helion in November, and therefore more likely to be projected 
on the sun's disk. After the lapse of ages, the months will 
change, on account of the slow retrograde motion of the nodes. 

27 5. Intervals between transits. — -While the earth makes 
13 revolutions from a node to the same node again, Mercury 
makes 54 revolutions, very nearly. Hence, in 13 years after a 
transit, the two bodies will return so nearly to the same rela- 
tions to the node, that another transit is likely to occur. The 
least interval between transits at the same node is 7 years, in 
which time Mercury makes very nearly 29 revolutions. As 
these are both odd numbers, the period may be halved, and a 
transit may occur in 3J years at the other node. This is the 
shortest interval. The transits of Mercury in the last half of 
the present century are the following: November 11, 1861; 
November 4, 1868 ; May 6, 1878 ; November 7, 1881 ; May 
9, 1891 ; November 10, 1894. 

VENUS. 

276. Tabular statements. — Mean distance from the sun, 
66,822,000 miles ; periodic time, 7J months ; diameter, 7,660 
miles ; diurnal rotation, 23.35 hours ; specific gravity, 4.S. 

277. Apparent motions. — Like Mercury, Yenus appears to 
pass back and forth by the sun, reaching a distance of 47° at 
its greatest elongation. This proves it to be an inferior planet, 
between Mercury and the earth. Its sidereal period approaches 
60 near to that of the earth, that its synodic period is length 

10 



146 TEANSITS OF VENUS. 

ened to nearly If years. Hence, after making an apparent 
retrograde motion, as LM (Fig. 69), it advances twice and two- 
thirds round the heavens before it commences the next retro- 
grade arc, OP. 

278. Phases and brightness of Venus. — Yenus passes 
through the same changes of phase as Mercury. But its ap- 
parent diameter, when the crescent phase is narrowest, is more 
than 6 times as great as when at full. For its distance from 
as, in the former case, is 92,000,000 — 67,000,000 == 25,000,000 
miles ; and in the latter, it is 92,000,000 + 67,000,000 = 
159,000,000 miles, a distance more than six times as great as 
the other. 

Yenus is the brightest of the planets, and has been known 
from ancient times as the morning and evening star, according 
as it is west of the sun, or east of it. 

The place of greatest brightness for Yenus is when about 40° 
from the sun, between the point of greatest elongation and the 
inferior conjunction. In this situation, it is frequently visible 
all day. 

279. Transits of Venus. — The orbit of Yenus is inclined 
to the ecliptic about 3 J degrees. The sun passes its nodes in 
June and December; therefore, the transits of that planet 
occur in those months. 

Yenus makes 13 revolutions in very nearly the same time in 
which the earth makes 8. Hence, a transit of Yenus at either 
node is usually preceded or followed by another at the same 
node, at an interval of 8 years. But this interval can not be 
halved, as in the case of Mercury (Art. 275), to find the time 
of a transit at the other node ; because, 8 being an even, and 
13 an odd number, there would, in 4 revolutions of the earth, 
be 6^ revolutions of Yenus, which would bring the two planets 
on opposite sides of the sun. 

The interval of 235 years is much more exactly measured by 
382 revolutions of Yenus. Therefore, after a transit, there is 
almost a certainty of another at the same node in 235 years. 
But, for the same reason as before, the middle of this interval 
can not be taken as the date of a transit at the other node. 



PARALLAX OF THE SUN. 147 

The smaller intervals must be obtained by using the period of 
227 years, which is 8 years less than 235 years. 

In 227 years, there are 369 revolutions of Yenus within 
1J days. Hence, transits are very likely to occur at the same 
node at intervals of 227 years. And at the middle of this in- 
terval, there will probably be a transit at the other node, since 
113^ revolutions of the earth, and 184 J of Yenus, bring both 
bodies to the opposite side of the heavens. This interval oi 
113^ years may be increased or diminished by 8, to furnish two 
other intervals. Hence, the ordinary intervals are 8, 105^ r 
113-J, and 121^ years, as may be seen in the following series of 
transits from 1518 to 2004: 

Interval. 

June 5th, 1518 

June 2d, 1526 8 years. 

Dec. 7th, 1631 1051 « 

Dec. 4th, 1639 ....... 8 

June 5th, 1761 1211 " 

June 3d, 1769 ....... 8 

Dec. 8th, 1874 105 J " 

Dec. 6th, 1882 8 

June 7th, 2004 121J « 

280. Parallax of the sun by a transit of Venus. — The 
planet Yenus is so near the earth, that its transit across the 
sun's disk is peculiarly favorable for obtaining the sun's paral- 
lax. Let E (Fig. 70) be the earth, Y Yenus, and fde the disk 
of the sun. Suppose observers stationed at A and B, the ex- 
tremities of that diameter which is perpendicular to the orbit 
of Yenus. Each one sees the planet describe a chord across 
the sun's disk from east to west. A observes it to come on at 
c, and leave at d ; while to the view of B, it comes on at e, and 
leaves at/1 And when it appears at a to the former, it is seen 
at b by the latter. It is the distance between the two projec- 
tions at a and b which is to be determined. 

281. The length of ah in miles. — Since the periodic times 
of the earth and Yenus are known, the ratio of the distances of 
E and Y from the sun is also known, by Kepler's third law. 
Hence, by subtraction, the ratio of the lengths of the triangles 



148 EXTEENAL AND INTEENAL CONTACTS. 

VA and Ya is known. These triangles may be regarded aa 
isosceles ; therefore, as they have equal angles at Y, they are 
similar. Hence, YA : Ya : : AB : ab. Thus, from the known 
ratio of YA to Ya, and the length of AB, we have the length 
of ab in miles. 



Fig. 70. 




B 



282. The length of sib in seconds. — We next wish to obtain 
the angular length of ab. The observers carefully mark the 
moment of entering on the disk, and the moment of leaving it. 
Thus, the length of time occupied by the transit, as seen by 
each observer, is carefully obtained. But since the angular 
motion per hour, both of the planet and the sun, is known, the 
time of crossing the disk can be changed into an arc ; and we 
thus have the number of seconds of a degree in the chord cd, 
and also the number in ef, and, therefore, in their halves, ca 
and eh. But the number of seconds in the sun's semi-diameter, 
cS or dS, is known. Hence, in the right-angled triangles c&a, 
tSb, we readily find the seconds in Sa and 8b, the difference 
between which is the length of ab in seconds. Thus, we find 
what angle is subtended by a line of given length, when placed 
at the sun, and viewed from the earth ; or, which is the same 
thing, placed at the earth, and viewed from the sun. There- 
fore, we know what angle at the sun is subtended by the 
radius of the earth ; and that is the sun's horizontal parallax. 

283. External and internal contacts. — At inferior conjunc- 
tion, the planet Yenus subtends an angle of more than 1/, and 
therefore, in the transit, appears like a small black circle, whose 
diameter is f s of the sun's diameter. To observe the beginning 
and end of a transit, the instant of external contact must first 
be noted, and afterward, when the planet has come wholly 
upon the disk, the time of internal contact also. The mean ot 
these is the time at which the center crosses the edge of the 



APPARENT MOTIONS OF MARS. 149 

disk. The duration of the transit is the interval between the 
moments at which the center of the planet enters and leaves 
the disk. 

284. Situation of the observers. — The observers can not 
probably be at points diametrically opposite, nor can they re 
main stationary during the transit, on account of diurnal 
motion ; therefore, allowance must be made for these circum- 
stances. In order that several independent results may be 
obtained, many stations are chosen, at the greatest possible 
distance from each other. In the observations on the transit oi 
1769, one of a large number of stations was in Lapland, and 
another on one of the Sandwich Islands. The result arrived at 
was, that the sun's horizontal parallax is 8". 5776, which, how- 
ever, is now considered to be too small. (See Preface.) 

MARS. 

285. Tabular statements.— Mean, distance from the sun, 
140,760,000 miles ; periodic time, 2 years ; diameter, 4,211 
miles; diurnal rotation, 24.62 hours; specific gravity, 4.17. 

286. Situation of Mars in the solar system. — This is the 
most remote planet of the first group described in Art. 256 — 
namely, Mercury, Yenus, Earth, Mars. It is also the nearest 
to the earth of those planets which are called superior. 

As Mars revolves in an orbit outside of the earth's, it can 
come into opposition to the sun, as well as into conjunction 
with it, appearing at every degree of elongation from 0° to 
180°. 

287. Apparent motions. — The real motion of Mars is from 
west to east; and during most of the year, its apparent motion 
is in the same direction, sometimes accelerated, and sometimes 
retarded, by the earth's motion. Near opposition, however, 
when the earth overtakes and passes by Mars, its motion ap- 
pears retrograde. Thus, let the earth make one revolution 
from F to F again (Fig. 71), while Mars describes nearly a hall 
revolution from G to 1ST. When the earth is at F, Mars ap- 



150 



APPARENT MOTIONS OF MARS. 



pears in the direction FG ; when at A, Mars at H, appears iw 
the sky at O ; when the earth is at B, Mars at I, appears at P. 
Thus far, the motion has been in advance, though becoming 
retarded near P. But as the earth passes from B, through C r 
to D, Mars, passing over the shorter arc IKX, appears to retro- 
grade from P to Q ; after which it again advances, appearing 
at E when the earth is at E, and in the direction FN when 
the earth is at F. 



Fig. 71. 




For the same reason, all the superior planets have a retro- 
grade motion at the time of opposition. 

288. Phases, and changes of apparent size. — At opposition^ 
M (Fig. 72), and at conjunction, M', it is obvious that Mars 
appears full, since we look in the same direction in which the- 
sun shines upon it. In other positions, the angle between the 
sun's rays and our visual line is acute, and the phase is gibbous 
(Art. 170). The planet is so near us, that the phase differs 
perceptibly from the full, when about half-way from conjunc- 
tion to opposition, as at Q, Q'. 

At opposition, Mars is nearer to us than at conjunction by 
the diameter of the earth's orbit. This makes its mean distance 
at opposition 48,000,000 miles, and at conjunction, 233,000,000. 
But on account of the elliptical form of both orbits, the 
least distance is 34,000,000 miles, and the greatest, 247,000,000 
miles. 



APPEAKANCE OF DISK. 



151 



289. Orbit and equator of Mars, — The orbit of Mars is in- 
clined to the ecliptic nearly 2°, and has an eccentricity equal 
to T V 




In its diurnal rotation, it considerably resembles the earth, 
having about the same length of day, and its equator being in- 
clined nearly 29° to its orbit. Hence, the seasons vary some- 
what more than those on the earth. 



290. Appearance of dish. — Mars is remarkable among the 
planets for its redness. The telescope reveals some permanent 
inequalities of surface, by which its diurnal rotation has been 
determined more satisfactorily than in the cases of Mercury 
and Yenus. And there are other appearances, which change 
as the relation of the equator to the sun changes. The polar 
regions, when turned away from the sun, exhibit a whiteness, 
which is supposed to be the effect of ice and snow ; and thia 
whiteness disappears gradually, when the pole is turned again 
toward the sun. 

290a. Satellites. — Mars has two satellites, discovered August 
11-17, 1877, by Prof. Asaph Hall, of Washington. They are 
remarkable for their small size, and for the proximity of the inner 
one to the planet ; being but 4,000 miles from its surface, and 
consequently revolving in the brief time of 7 hours, 38 minutes. 



152 CHARACTEEISTICS OF PLANETOIDS 



CHAPTEE XVI. 

THE PLANETOIDS. — JUPITER.— SATURN. — URANUS. — NEPTUNE. 

291. The space between the four s?n all planets and the four 
large ones. — The large interval between Mars and Jupiter, 
which seemed to break the continuity of the series of planets,, 
was noticed by Kepler. About the close of the last century, 
Bode, of Berlin, showed that a series of numbers, following a 
certain law, would express pretty accurately the planetary dis- 
tances from the sun, if only the vacancy between Mars and 
Jupiter were supplied. This led to a special search for new 
planets, which was presently rewarded by the discovery of sev- 
eral small bodies, which have been called asteroids, planetoids, 
or minor planets. 

THE PLANETOIDS. 

292. Their number, and the time of their discovery. — Four 
of these bodies were discovered within the first seven years of 
the present century — namely : Ceres, Pallas, Juno, and Yesta. 
Since 1845, others have been found nearly every year, till their 
number at the present time (1884) is over two hundred. The 
whole number of planetoids may be regarded as indefinitely 
great. 

293. Characteristics. — They are distinguished from the 
eight planets in the following particulars : 

1. By their diminutive size. — They are invisible to the 
naked eye, and by the telescope can not be distinguished from 
faint fixed stars, except by their motion. They are generally 
too small to show a sensible disk, and hence can not be meas- 
ured with any certainty. The largest of them is believed to be 
only about 200 miles in diameter. And it is estimated by the 
slight disturbing influence which they exert, that their entire 
mass is equal only to a small fraction of the earth. 



jupiter's magnitude. 153 

2. By the large eccentricity and obliquity of their orbits. — 
The eccentricity of most of them is much greater than that 01 
any of the eight planets. 

The obliquity of the orbit of Hebe is 14°, and that of Pallas 
is 34°, which is the greatest yet discovered. 

3. By their being clustered in a ring. — The orbits vary con- 
siderably in size, and therefore the periodic times are various 
But as they are generally quite eccentric, nearly every planet- 
oid is nearer the sun at perihelion, than the others at aphelion. 
The orbits are therefore all linked together, and pass through 
each other. Thus, the planetoids are to be regarded as moving 
among each other about the sun, within the limits of a ring, 
whose breadth, in the direction of the radius vector, is more 
than 160,000,000 miles. Flora, which moves in the smallest 
orbit yet discovered, performs its revolution in 3-J years ; 
Hilda, the most remote, in 8 years. Their mean periodic time 
is 4J years ; and their mean distance from the sun is 250,000,000 
miles. 

294. Modes of designating them, — Feminine mythological 
names have been applied to all the planetoids which have yet 
been discovered. But the more convenient method, and the 
one most used, is to express each planetoid by a number, show- 
ing its place in the order of discovery, this number being in- 
closed in a circle, which indicates a disk. Thus, Ceres is (T) ; 
Thetis, @; Pandora, @; etc. See Table V., at the end. 

JUPITER. 

295. Tabular statements. — Mean distance from the sun, 
480,638,000 miles ; periodic time, 12 years ; diameter, 86,657 
miles ; diurnal rotation, 9.92 hours ; specific gravity, 1.3. 

296. Jupiter's magnitude and place in the solar system. — 
Jupiter is the nearest of the large planets outside of the planet- 
oids, and its orbit is not far from 130,000,000 miles beyond the 
ring which includes them. On account of its great distance 
from the sun, compared with the earth's, Jupiter presents to us 
no visible change of phase, appearing always full. Its disk, a& 



154 THE BELTS OF JUPITER. 

presented to us, is almost the same as if we were at the sun 
The same is, of course, true of all the planets still more re* 
mote. 

Jupiter greatly surpasses all the other planets in magnitude. 
In volume, it is about 1| times the sum of all the others, and in 
mass, more than 2£ times their united mass. 

297. Its spheroidal form. — Though the diameter of Jupiter 
is 11 times that of the earth, yet it rotates on its axis in less 
than 10 hours ; so that the equatorial velocity is about 27 times 
as great as the earth's. This rapidity of rotation produces a 
sensible oblateness of the planet. Its ellipticity is ^ Y ; and so 
considerable a deviation from the spherical form is perceptible 
to the eye without measurement. 

298. The belts of Jupiter. — This name is given to bands or 
stripes of darker shade than the rest of the disk, stretching 
across it in the direction of its rotation (Fig. 4, Fr.) They vary 
from time to time in number and in breadth, often covering a 
large part of the surface. A belt usually appears of uniform 
breadth entirely across, but not always ; its edge is occasionally 
broken, and sometimes it is much wider on one part of the 
disk than on the other, the change of breadth being commonly 
quite abrupt, and thereby revealing the rotation of the planet. 
There are, ordinarily, two conspicuous belts, lying near the 
equator, one north, and the other south of it. 

299. Supposed cause of the belts. — The belts are considered 
as affording proof that Jupiter is surrounded by an atmosphere, 
in which clouds are floating. As a consequence of the exceed- 
ingly rapid rotation of the planet, there would be very power- 
ful currents, analogous to the trade-winds of the earth ; and 
the clouds would be thrown into the form and arrangement of 
zones parallel to the equator. The clouds would reflect the 
sun's light to us more strongly than the atmosphere ; and the 
dark belts, therefore, are the unclouded portions, through which 
we look on the body of the planet. 

300. Orbit and equator of Jupiter. — The orbit <rf fupitej 



satellites of jupitek. 155 

is nearly coincident with the plane of the ecliptic, its inclina- 
tion being only 1° 19'. Its eccentricity is ^V? which is three 
times as great as that of the earth's orbit. 

The equator of Jupiter is inclined a little more than 3° to 
its orbit. There is, therefore, no perceptible change of seasons 
on that planet. 

301. Satellites of Jupiter. — These are four in number, re- 
volving in orbits very nearly circular, and in planes which 
make small angles, both with the orbit and the ecliptic. They 
are called the first, second, third, and fourth, reckoning out- 
ward from the planet. 

302. Their revolutions. — On account of the position of the 
orbits, we see the satellites passing back and forth across the 
place of Jupiter, nearly in straight lines (Fig. 4, Fr.) From 
their greatest elongation west of Jupiter, they advance to the 
greatest elongation on the east, passing behind the planet on 
their way. Then, after remaining stationary a short time, they 
retrograde to the west side, passing between us and the planet. 
These movements prove that they revolve from west to east, as 
all the primary planets do. At the greatest elongation on 
the east side, they are, for a little while, stationary, because 
coming toward us ; and on the west side also, because going 
from us. 

303. Their size, distance, and periods. — The diameters of 
the first, third, and fourth satellites are greater than that of the 
earth's moon, but the diameter of the second is a few miles 
less. To us, they appear as stars of the 6th or 7th magni- 
tude ; but on account of the brightness of the primary, they 
can very rarely, if ever, be seen by the naked eye. If two or 
three satellites happen to appear very near together, they may 
possibly be seen by the naked eye, when they of course seem 
to be one. The first is further from Jupiter than the moon is 
from the earth, and the fourth nearly five times as far. Their 
periods of revolution are very short, compared with the moon's ; 
for, on account of the strong attraction of Jupiter, great velocity 
is requisite to maintain them in their orbits. 



156 



ECLIPSES AND OCCULTATIONS. 



Satellites. 


Diameters. 


Distances. 


Sidereal Revolutions. 


1 

2 
3 
4 


2,365 
2,123 
3,471 

2,966 


260',370 

414,360 

660,900 

1,162,400 


1 d. 18 h. 28 m. 
3 " 13 " 15 " 
7 " 3 " 43 " 
16 " 16 " 32 " 



304. Their configurations. — Tlie relative positions of Jupi 
ter and the four satellites, as seen from the earth, are inces- 
santly varying. We most frequently see two or three on one 
side, and two or one on the other; rarely all on one Fide. 
Very often, one or two are invisible, being either behind 
Jupiter, or projected on it. Sometimes, three are thus con- 
cealed, and in very rare instances, all four. 

305. Eclipses and occultations of Jupiter and its satel- 
lites. — The great dimensions of Jupiter and its shadow, and 
the small inclinations between the ecliptic, Jupiter's orbit, and 
those of its satellites, cause very frequent eclipses and occulta- 
tions. A. satellite of Jupiter is eclipsed when it goes through 
the shadow of the planet ; it suffers occultation when it is 
hidden from our view by passing behind the planet. The first, 
second, and third satellites pass through both eclipse and oc- 
cultation at every revolution, and the fourth rarely escapes. 

Besides these two classes of phenomena, there are two others, 
— namely, the eclipse of Jupiter, when its satellite casts a 
shadow upon it ; and an occultation of Jupiter, when a satel- 
lite passes between it and the earth. The eclipse is a small 
black spot passing over the disk. The occultation is scarcely 
perceptible, because the planet and satellite are of about equal 
brightness. On a belt, the satellite may appear brighter 
and between two belts it may appear less bright than the 
primary. 

306. Order of eclipses and occultations. — "When Jupiter is 
east of opposition, the eclipse always precedes the occultation ; 
when west of opposition, the occultation precedes the eclipse. 
For, let S (Fig. 73) be the sun ; A, B, C, several positions of 
the earth ; J, Jupiter ; and EHK, the orbit of a satellite. The 



ECLIPSES AND OCCULTATIONS. 



157 



Oodies are supposed to revolve in the order of the letters. Ii 
the earth is at A, SA produced marks the place of opposition, 
and Jupiter is east of that place. The satellite enters the 
shadow at E, emerges at F, and then passes behind the planet 
at G-, and reappears at H. In this case, the eclipse is past be- 
fore the occultation begins. In the same manner, the eclipse 
of Jupiter begins when the satellite is at K, and ends when at 
L ; and the occultation follows it, while the satellite moves 
from M to N". If the earth were at C, Jupiter would be west 
of opposition — that is, west of SC produced. And it is obvious 
that the satellite would go behind the planet before entering 
the shadow, and also would appear between us and the planet 
before casting a shadow on it. 



Fig. 73. 




The earth is not, in general, so situated that one phenom- 
enon is closed before the next begins; and it is never true oi 
the first satellite. The case is represented by the orbit eKkn. 
The eclipse begins at e, and the occultation ends at h; but the 
end of the eclipse and the beginning of the occultation are not 
seen. In the same manner, the eclipse of Jupiter begins at yfc, 
and the occultation ends at n. During a part of the interven- 
ing time, the shadow and the body of the satellite are both 
seen, projected at different places on the primary. 

At the time of opposition, the earth being at B, the eclipse 



158 Saturn's disk. 

s>f a satellite obviously occurs entirely within its occultation, 
and the occultation of Jupiter entirely within its eclipse. 

It is found that there exists such a relation between the 
mean motions of the three first satellites, that they can never 
all be eclipsed at the same time. 

307. The velocity of light discovered by the eclipses oj 
Jupiter's satellites. — In 1675, it was discovered by Roemer 
that eclipses occurred earlier than the calculated time, when 
the earth is in that part of its orbit which is near to Jupiter, 
and later, when in the remote part. The eclipses of any one 
satellite are so frequent, that the mean interval between them 
is obtained with great accuracy ; and by this mean interval, 
the times of future eclipses could be calculated. But it was 
perceived that while the earth moves from the remote side to 
the nearer side of its orbit, the real intervals are shorter than 
the mean, so that, at the nearest point, an eclipse occurs about 
8m. 13-^s. too soon. Again, as the earth goes to the side of its 
orbit furthest from Jupiter, the real intervals are all greater 
than the mean ; and at the most distant point, an eclipse is 
later than the calculated time by 8m. 13-Js. Roemer attributed 
this periodical error of time to the progress of light, and in- 
ferred that light requires 16m. 27s. to cross the earth's orbit. 
This makes the velocity of light near 187,000 miles per sec- 
ond ; which seemed at first quite incredible, and was received 
with distrust. But its correctness was soon established by the 
discovery of the aberration of the stars, which gives about the 
same result (Art. 146). 

SATURN. 

308. Tabular statements. — Mean distance from the sun, 
881,203,000 miles; periodic time, 29 years ; diameter, 70,500 
miles ; diurnal rotation, 10.21: hours ; specific gravity, 0.8. 

309. Saturn's disk. — Saturn is the second planet in size ; 
and being the second in order beyond the planetoids, is not too 
far from the earth to present a large disk. Its form is seen to 
be elliptical, and it is faintly striped with belts in the direction 



satukn's RItfGS. 159 

of the major axis. Both these appearances are explained by 
the rapid rotation of the planet on its axis, as in the case oi 
Jupiter. Its ellipticity is j 1 ^. 

310. Saturn's rings. — The distinguishing feature of this 
planet is the system of broad thin rings which surround it. 
They lie in a plane inclined about 2S° to the ecliptic, and 
therefore generally present an elliptical appearance to the 
earth (Fig. 3, Fr.) The ring, as usually seen, consists of two 
rings, the inner of which is the widest. The inner edge is 
19,000 miles from the surface of the planet ; and the diameter 
from outside to outside is 168,000 miles. The line in which 
the plane of the ring intersects the plane of Saturn's orbit is 
called the line of the nodes. 

Within the double ring already described, there is a much 
fainter one, which can not be seen with ordinary telescopes. 
By careful observations, it is also perceived that there are sev- 
eral concentric divisions of the rings, which vary their number 
and position from time to time. These fainter divisions are 
invisible, except at the ends of the ellipse. The rings lie in 
one plane, and are exceedingly thin. The latest measurements 
make their thickness less than 40 miles. A circle of common 
writing paper, one foot in diameter, would be too thick to rep- 
resent it correctly. But the thickness appears not to be uni- 
form ; for in the edge view, it often presents the aspect of a 
broken line, as though some parts were thick enough to be seen, 
and others not. There seems to be evidence that the rings 
consist either of liquid matter, or else of solid matter in a dis- 
integrated condition. 

311. Rotation of the rings. — Such rings of matter around 
Saturn could no more be sustained without rotation, than the 
moon could remain at its distance from the earth without re- 
volving about it. They are found to rotate in their own plane 
within the short period of 10J hours, nearly the same as the 
period of the planet itself. The outer edge of the ring must, 
therefore, have a velocity of 14 or 15 miles per second. 

312 • The plane of the rings' always parallel to itself. — 



160 Saturn's rings. 

During the revolution of Saturn around the sun, occupying 
about 29 years, the rings maintain everywhere the same posi- 
tion in relation to the plane of Saturn's orbit as represented in 
Fig. 74, in which ah is the earth's, and ACEG Saturn's orbit, 
seen obliquely. While the planet passes through the half revo- 
lution ACE, the north side of the rings is seen by an observer 
on the earth as an ellipse, more or less eccentric ; but during 
the other half, EGA, the south side is in view. Each of these 
periods occupies near 15 years. When Saturn is near A and 
E, the line of nodes passes across the earth's orbit, and the 
edge of the rings is therefore directed toward the sun and 
earth ; and at those times it fills too small an angle to be seen, 
except by the best instruments. 

Fig. 74. 




313. Passage of the plane of the rings across the earth? s 
orbit. — The motion of Saturn is so slow, that it requires almost 
a year for the plane of its rings to pass by the whole diameter 
of the earth's orbit. Let DF (Fig. 75) be the earth's orbit, and 
AC a portion of Saturn's. Suppose these orbits to lie in the 
plane of the paper, and the plane of the rings to be inclined 
about 28° to the paper, making the common section of the two 
planes in the lines AD, BG, etc. Saturn is 9.54 times as far 
from the sun as the earth is. Therefore, SA : SD : : 9.54 : 1 
: : rad. : sin SAD; .\ SAD, or its equal, ASB = 6° V ; .-, ASC 
= 12° 2'. Knowing Saturn's periodic time, we readily find 
that it will describe 12° 2' in 359^ days, near six days less than 
a year. Hence, while Saturn passes from A to C, the earth 
will pass very nearly around its orbit, DEFG. But the earth 



DISAPPEARANCE OF THE RINGS. 



161 



may be at any point of its orbit when the planet reaches A. 
The disappearances of the rings will vary according to the 
positions of the earth. 

Fig. 75. 




314. Circumstances of the disappearances. — There are three 
ways in which the rings may fail to be visible during the 
period in which the line of their nodes is crossing the earth's 
orbit. 

1. The ring may present its edge exactly to the earth, when, 
in common telescopes, it subtends too small an angle to be 
seen. 

2. It may present its edge exactly to the sun, so that neither 
side of the ring is enlightened. 

3. Its plane may be directed between the earth and sun, when 
the dark side is toward us. 

The disappearance by either of the two first causes may be 
considered as only momentary; for the line of nodes passes 
the breadth of the sun in less than 2 days, and of the earth in 
about 20 minutes. But the third cause may conceal the ring 
from our view for weeks or months. This prolonged disappear- 
ance may occur either once or twice, or possibly not at all, while 
the line of nodes is passing the breadth of the earth's orbit. 

315. One disappearance. — If the earth is at F when the 
planet reaches A, then the earth will go from F nearly to D, 
while the nodal line advances from AD to BS, and the earth 
will pass the line between G and D, as at K. Up to that 
point, the luminous side is presented toward the earth ; but 
from K to a point near D, the plane of the rings falls between 
the earth and sun, and the rings are invisible, and continue so 

11 



162 DISAPPEARANCE OF THE RINGS. 

about two months. When the nodal line has passed the son, 
the luminous side of the rings is again toward the earth ; and 
before the earth completes the half orbit DEF, the nodal line 
will pass off at F. 

316. Ttvo disappearances. — If the earth has advanced some 
distance on the quadrant FG — for example, to the middle L — • 
when the nodal line touches D, then the earth passes the lint, 
between K and D, and the dark side is toward us. The line 
passes the sun when the earth is near the middle of DE, after 
which, the rings are seen. But before the nodal line reaches 
CF, the earth will overtake it, and be on the dark side again. 
Between F and L, the earth once more crosses the line, and the 
rings present to us their bright side. In this case, the rings 
disappear twice during the nodal year. 

These two periods of disappearance may be so prolonged as 
to unite in one of about eight months in length. This happens 
when the earth is two' or three days past G, at the time when 
the nodal line touches D. Then, before reaching D, the earth 
passes to the dark side of the rings, and continues on that side 
till both the earth and the nodal line pass E together. As 
soon as that point has been passed, the line is again between 
the sun and earth, and continues so mi til it is recrossed by the 
earth on the quadrant FG. 

317. No disappearance. — It is possible that no disappear 
ance, which has continuance, should happen during the nodal 
year. Suppose the earth two or three days past E, when the 
line of nodes reaches D. Then, while the line moves from AD 
to BS, the earth will advance to G, all the time on the lumi- 
nous side of the rings ; the earth and sun will both be in the 
line BSG at once, the planet being in conjunction ; and after 
the earth has passed G toward D, the bright side of the rings 
is in view, as before, and will continue so. Thus, there is only 
a momentary disappearance, and that, when the planet and 
rings are lost in the blaze of the sun's light. 

In general, there are two periods of disappearance within 
the nodal year, arising from the third cause, each beginning 
and ending with a disappearance from the first or second cause. 



DISCOVERY AND PLACE OF URANUS. 163 

318. Phenomena of the rings at the planet. — On that 
hemisphere of the planet to which the luminous side of the 
rings is presented, there is the appearance of splendid arches 
spanning the sky, having a breadth and elevation according to 
the latitude of the place. At latitude 30°, the breadth is about 
18°, and the elevation of the lower edge on the meridian about 
22°. E"ear the poles, however, it is below the horizon. The 
luminous side is presented to the northern hemisphere near 15 
years, and then the same length of time to the southern hemi- 
sphere, in regular alternation. 

A part of the rings is generally eclipsed by the shadow of the 
planet falling on it. 

Also, during the 15 years in which the dark side of the 
rings is turned toward a hemisphere, its shadow is cast across 
a zone of it, which causes an eclipse of the sun. And at a 
given place, a total solar eclipse may continue from day to day, 
without interruption, for several years. 

319. Satellites of Saturn. — Saturn is attended by eight 
satellites. Their periods of revolution vary from less than one 
day to 79 days. Their diameters vary from 500 to 2 3 900 
miles ; but on account of their immense distance from the 
earth, they are seen only with the best instruments. They are 
all external to the rings, at distances from the planet, varying 
from 122,000 to 2,338,000 miles. Their orbits are nearly in 
the plane of the rings, and make an angle of about 28° with 
the orbit of the planet. Hence, they are not very liable to be 
eclipsed. The principal time for eclipses is that at which the 
rings disappear ; for then the sun is nearly in the plane of their 
orbits, as well as of the rings. 

URANUS. 

320. Tabular statements. — Mean distance from the sun, 
1,772,088,000 miles ; periodic time, 84 years; diameter, 31,700 
miles ; specific gravity, 1.3. 

321. Discovery, and place in the system. — Uranus was un- 
known to the ancient astronomers; and to them, therefore, 



164 DISCO VEEY OF NEPTUNE. 

Saturn's orbit was the boundary of the solar system. Uranus 
was discovered by Sir William Herschel, in 1781, and has 
made but little more than one revolution since that time. It 
was, however, repeatedly seen by earlier astronomers, and re- 
corded in their catalogues as a fixed star. By this discovery, 
the diameter of the known solar system was doubled. 

Uranus is the third of the four great planets in order of dis- 
tance, but it is least in diameter. Its distance from us is so im- 
mense that it appears only as a faint star, and presents no in- 
equalities by which its diurnal motion can be discovered. Its 
orbit is very nearly circular, and is inclined less than a degree 
to the ecliptic. 

322. The satellites of Uranus. — Sir William Herschel an- 
nounced the discovery of six satellites belonging to Uranus. 
But only four have been identified by later astronomers. The 
remarkable facts relating to these satellites are, that their orbits 
are nearly at right angles to the plane of the ecliptic, and that 
in the orbits, the motions of the satellites are retrograde — that 
is, from east to west. Their periods of revolution vary from 
2i days to 13J days, and their distances from 123,000 to 
376,000 miles. 

NEPTUNE. 

323. Tabular statements. — Mean distance from the sun, 

2,777,948,000 miles; periodic time, 165 years; diameter, 
34,500 miles ; specific gravity, 1.1. 

324. Discovery. — Neptune was discovered in 1846. The 
circumstances which led to the discovery were briefly as fol- 
lows. After the orbit of Uranus had been carefully computed, 
and corrections made for the disturbing influence of Jupiter 
and Saturn, the planet was found to depart from the calculated 
path in a manner not to be accounted for, except by suppos- 
ing some other disturbing force. It was for some time sus- 
pected that there must be a planet superior to Uranus, whose 
attraction caused the change of its orbit. At length, two 
mathematicians, Le Verrier, of France, and Adams, of England, 



ELEMENTS OF ORBITS. .165 

*jaeh without any knowledge of what the other was attempting, 
engaged in the arduous labor of calculating what must be the 
elements of a planet which should produce the given disturb- 
ance of the motions of Uranus. They reached results which 
agreed remarkably with each other. Le Yerrier communicated 
to Galle, of the Berlin observatory, the place in the sky in 
which the disturbing body should be situated ; and in the 
evening of the same day, Galle found it within a degree of the 
predicted longitude. 

The planet thus discovered explains fully the disturbances in 
the motions of Uranus. 

It soon appeared that Neptune had repeatedly been entered 
in catalogues as a fixed star. The earliest of these records, in 
1795, afforded material aid at once in determining its mean 
distance and its periodic time. 

Neptune is attended by one satellite, which was also dis 
covered in 1846. It is nearly as far from the primary as tli6 
moon is from the earth, and revolves in 5d. 21h. 



CHAPTER XVn. 



ELEMENTS OF A PLANETARY ORBIT. — QUANTITY OF MATTER 
IN THE SUN AND PLANETS. — PLANETARY PERTURBATIONS. — 
RELATIONS OF PLANETARY MOTIONS. 

325. Elements of an orbit. — These are the quantities which 
must be known, in order to calculate the place of a planet at a 
given time. They are seven in number. 

1. The periodic time. 

2. The mean distance from the sun, or the semi-major axis 
of the orbit. 

3. The longitude of the ascending node. 

4. The inclination of the plane of the orbit to that of the 
ecliptic. 

5. The eccentricity of the orbit. 

6. The longitude of the perihelion. 



166 PERIODIC TIME. 

7. The place of the planet in its orbit at a given epoch. 

Two of these, 3d and 4th, determine the position of the plane 
in which the orbit lies ; the second fixes the size of the orbit ; 
the 5th, its form ; the 6th, the relation of the form to the plane 
of the ecliptic; the 1st and 7th, the circumstances of the 
planet's motion in the orbit. 

The orbit of a planet can not be determined by the same 
method as the moon's orbit is (Chap. X.), or the sun's apparent 
orbit (Chap. IV.), because it is not the earth, but the sun, whick 
occupies the center of the planetary revolutions. 

326. Geocentric and heliocentric place of a planet. — The 
point in the celestial sphere which a planet occupies, as seen 
from the earth, is called its geocentric place ; its place as seen 
from the sun is called its heliocentric place. It has already 
been noticed that the planets, as seen from the earth, have a 
retrograde motion during a part of every synodical revolution. 
This is the effect of the observer's position and motion, and would 
not exist if he were stationed at the sun. The place of a planet, 
as seen from the earth and the sun, can never agree, except 
when the sun and earth are on the same side of the planet, and 
in the same straight line with it. But after the relations of the 
earth to the planet and to the sun are obtained, there is no dif- 
ficulty in calculating the heliocentric place of the planet. 

327. First element — the periodic time. — This is found by 
observing the time that intervenes between the two successive 
returns of the planet to the same node. 

It may be known when a planet is at a node, because then 
its latitude is nothing. If, from a series of observations on the 
right ascension and declination of a planet, the latitudes are 
computed, and one of them is zero, then the exact time of pass- 
ing the node is obtained. But if, as is usually the case, the 
two least latitudes are, one north, and the other south, the time 
of passing the node between them is readily found by a propor- 
tion. Similar observations are made when the planet again 
arrives at the same node, and thus the periodic time becomes 
known. 

It is discovered that a minute correction of the "oeriodic time. 



DISTANCE FROM THE SUN. 167 

thus derived, must be applied for the retrograde motion of the 
node. The periodic time of a planet may also be derived frorc 
the observed length of its synodic revolution — that is, the inter 
val between two successive oppositions, or two conjunctions oi 
the same kind. The computation is similar to that employed 
in finding the sidereal period of the moon from its synodical 
period (Art. 158). 

In both the above methods, great advantage, in point of ac- 
curacy, is gained, if two very distant epochs can be brought 
into comparison, such as two distant passages of the node, or oi 
opposition. For example, a transit of Mercury occurs at in- 
ferior conjunction. Divide the interval between two observed 
transits, several years apart, by the number of synodical revo- 
lutions of Mercury which intervene, and its mean synodical 
period is very accurately obtained. 

328. Second element — the distance from the sun. — The dis 
tance of an inferior planet from the sun is found as follows 
Let S (Fig. 76) be the sun. E the earth, and C 
the planet. Measure the greatest elongation, 
SEC ; then, in the right-angled triangle, rad : 
sin SEC : : SE : SC. If the orbit is elliptical, 
the value of SC, as obtained at different times, 
will be different; and a great number of such 
observations should be made, in order to obtain 
the mean distance. 

The distance of a superior planet may be 
found by observations on its retrograde motion 
at the time of opposition. For, the more dis- 
tant the planet, the less will the earth's motion 
throw it, apparently, backward. Let S (Fig. 77) 
be the sun, E the earth, and M a superior planet. 
Let E pass over ~Ee in a short time, as one day, and let M pass 
over Mm in the same time. As the periodic times of E and M 
are supposed to be known, the angles ES<? and MSm are 
known, and, therefore, their difference, eSm, Join em, and 
produce it to X in SM produced. Draw ey parallel to SX : 
the angle ~K.ey is the retrogradation during the day in which 
the planets describe the arcs Ee and Mm, and is known by ob- 
servation. But SXd = X.ey ; and, therefore, in the triangle 




163 



LONGITUDE OF THE NODE. 



eSX, the third angle, XeS, is known. Hence, in the triangle 
Sem we have all the angles, and the side Se, by which 8m ia 
easily computed. This process may be repeated at every oppo- 
sition, and thus the mean distance is ultimately obtained. 




329. Third element — longitude of the node. — Let S (Fig. 
78) be the sun, EFG the earth's orbit, OPQ the orbit of a 
planet, CL an arc in the plane of 
the ecliptic, intersecting the orbit **■ 8 ' 

in P. SP is, therefore, the line of 
the nodes. And let EA, FA', and 
SA" be parallel lines, directed to- 
ward the vernal equinox. When 
the earth is at E, suppose the plan- 
et is at the node P ; then E, P, and 
S are all in the plane of the eclip- 
tic, and AEP is the longitude of P, 
and AES that of the sun. These 
longitudes being obtained, their dif- 
ference is SEP, which is, therefore, 
known. After the planet has per- 
formed a revolution to the same 
node again, suppose the earth to be 
at F; then we find, as before, its 
longitude, ATP, that of the sun, 
A'FS, and their difference, SFP. 
As the times are known in which 
the earth is at E and at F, we know SE, SF, and the angle ESF, 
and can compute EF, and the angles SEF and SFE. From 
the several angles at E and F, thus obtained, we derive PEF 
and PFE ; and these, with the side EF, give us the side FP. 
Then, in the triangle SFP, from SF, FP, and SFP, compute 
the angle FSP. From this, subtract A"SF (the supplement of 
A'FS), and there remains A"SP, the heliocentric longitude of 
the node. 

It is by processes of this kind that the slow retrograde mo- 




INCLINATION TO THE ECLIPTIC. 169 

tion of the nodes is discovered (Art. 327). It amounts to only 
a few minutes in a century. 

330. Fourth element — inclination of the orbit to the eclip- 
tic. — Select the time of observation, when the sun's longitude, 
obtained from the tables, is the same as the heliocentric longi- 
tude of the node ; and find for that time the geocentric longi- 
tude and latitude of the planet. Let E (Fig. 79) be the earth 
S the sun, P the planet, JNO the line of the nodes coinciding 
with ES ; and let EA and SA' be the direction of the vernal 
equinox. Join EP, and, with it as a radius, describe the sur- 
face of a sphere, cutting the plane of the ecliptic in the arc BC. 
From P draw the arc PQ, perpendicular to BC. AEO is the 
longitude of the sun, and A'SO, its equal, is the heliocentric 
longitude of the node O. AEQ is the geocentric longitude ot 
the planet. In the spherical triangle BPQ, right-angled at Q, 
PQ measures the given latitude, BQ measures the difference 
between AEQ and AES, and PBQ is the inclination to be 
found. Then, rad x sin BQ = tan PQ x cot PBQ ; 

. ™+ p-ro - rad • s in B Q . 
,\ cot r±>Q = — dt=t— ? 

tan PQ 
and the inclination of the orbit to the ecliptic becomes known. 

Fig. 79. 




331. To find the heliocentric longitude and latitude of a 
planet. — Let S (Fig. 80) be the sun, E the earth, EBC its orbit, 
P the planet, EA, SA 7 the direction of the vernal equinox. 
Let PQ be drawn perpendicular to the plane of the ecliptic 
AEQ is the geocentric longitude of the planet, A'SQ its helio- 



170 



ECCENTRICITY OF THE OEBIT. 



centric longitude. Also, PEQ is its geocentric latitude, and 
PSQ its heliocentric latitude. SEP, the elongation of the 
planet from the sun, is known from observation; SE, the 
radius vector of the earth's orbit, and SP, that of the planet's 
orbit, are also known. Therefore, PE may be computed. 
Knowing PE, and PEQ in the right- angled triangle, we can 
compute EQ. Then, in the triangle QES, EQ, ES, and th( 
angle QES (= AES - AEQ) being known, QSE and QS are 
found. From QSE, subtract ESA' (the supplement of AES), 
and A' SQ is obtained, which is the heliocentric longitude of P. 
Again, in the right-angled triangle PSQ, having SQ and SP, 
we find the angle PSQ, the heliocentric latitude. 

Fig. 80. 




332. Fifth and sixth elements — eccentricity of the orbit, 
and longitude of the perihelion. — A focus and three points in 
the curve of a conic section being given, its directrix can be 
determined, and the curve drawn. (Coffin's Con. Sec, Prop. 
II.) Tims, let SM, SjST, and SP (Fig. 81) be three radii vec- 
tores of an orbit, determined in length and position by the pro- 
cesses already described. If MN and NP are joined, the tri 
angles MNS and NPS are known in all respects. Then, if 
MN be produced, so that NK : MR : : NS : MS, E is a point 
of the directrix. Another point, L, is fixed in a similar man- 
ner by producing PN. The directrix being thus determined, 
draw perpendiculars to it from S, M, 1ST, and P. The axis ol 
the orbit is on KS produced. The ratio, SM : MG, is constant 
for every point of the curves (Cof. Con. Sec, Pr. II.} 



MASSES OF BODIES COMPAKED, 
Fief. 81. 



173 




The distance SK of the focus from the directrix is found 
thus. Draw MD perpendicular to SK. LNS, the external 
angle of the triangle NSP being known, subtract MNS from 
it, and we have LNB, and the including sides LIST, "KR, to find 
the angle B. This, with the side ME, in the right-angled tri- 
angle MGE, gives us GM, and the angle GME. Then, 
180° - (GME + EMS) - MSD, from which, and MS, we 
compute DS ; and GM + DS = SK. 

Tc find the perihelion, divide SK, so that SA : AK : : SM , 
MG ; A is the perihelion. 

To find the aphelion, produce KS to B, so as to make SB : 
BK : : SM : MG ; B is the aphelion. 

Bisect AB in C ; then SC divided by AC is the eccentricity 
of the orbit. 

The longitude of the perihelion is known from the angle 
MSA, already obtained ; for the longitude of SM is given at 
the outset. 



333. Masses of bodies compared by the orbits described 
about them. — The mass of a body, whether the sun or a planet, 
can be compared with that of another, by means of the distance 
and period of a planet or satellite revolving about each. It 
has been proved (Chap. VIII.) that gravity varies directly as 
the mass, and inversely as the square of the distance ; that is, 

G co y~-. It was shown, also (Chap. VI.), that the centripetal 



172 MASSES OF BODIES COMPARED. 

force or gravity varies directly as the distance, and inversely 
as the square of the periodic time ; that is, G- oo — . There- 

M D D 3 

fore, -^— go — -; .•. M qo — ; or, the united mass of the cen- 
tral and the revolving body varies directly as the cube of their 
distance apart, and inversely as the square of the periodic time. 
Thus, to compare the mass of the sun, about which the earth 
revolves, and the mass of the earth, about which the moon 
revolves, we have 

(238,820) 3 (92,381,000) 3 , QO , AAA , 

127T32F : (365.256)' '' ' ' ' J ' 

Therefore, the mass of the sun is about 324,000 times that 
of both earth and moon, or 327,000 times that of the earth alone. 

334. Examples. — 

1. Were the earth's mass equal to the sun's, in what time 
would the moon, at its present distance, revolve about it ? 
Letting x stand for the time required, we have 1 : 327,000 : ; 

X. Ans. lh. 8m. 48s. 



(27.32/ ' x l 

2. How much must the mass of the earth be increased, 
in order that the moon may revolve about it in the same 
time as it now does, when removed to three times its present 
distance ? Ans. It must be 27 times as great. 

3. The distance of Jupiter from the sun is 481,000,000 
miles, and its periodic time is 4332.554 days. The fourth 
satellite is 1,162,000 miles from the primary, and revolves 
in 16d. 16h. 32m. Compare the mass of the sun with that of 
Jupiter. Ana. 1048 : 1. 

4. The moon revolves in 27.32 days, at the distance of 
238,820 miles from the earth : Jupiter's second satellite re- 
volves in 3.552 days, at the distance of 414,000 miles. What 
are the relative masses of the earth and Jupiter ? 

Ans. 1 : 312. 

335. Masses of planets which have no satellites. — The 
method described in the preceding article can be applied to 



PERTURBATIONS OF THE PLANETS. 173 

the sun, and all those planets which are attended by satellites. 
But Mercury and Venus, which have no satellites, must 
be compared in some other way. Each of these planets, by its 
attraction, sensibly disturbs the motion of the planet nearest to 
it ; and the degree of this disturbance, the distance being 
known, is a measure of its quantity of matter. Thus, the 
masses of Venus and Mars can each be estimated, by observing 
the force which they exert on the earth when passing near it. 
The mass of Mercury has been determined by its disturbing 
power exerted on Eneke's comet, as well as on the planet 
Venus. 

336. Densities of the planets. — The masses of bodies vary 
as the products of their volumes and densities. Therefore 
their densities vary as the masses divided by the volumes. 
The densities, as given in Table IV, may be obtained in this 
way, and reduced to a scale, in which the earth's density is 
called 1. Or they may be reduced to a scale, in which the 
density of water is 1 ; in the last form, the numbers are called 
specific gravities. These also are given in Table IV 

337. Perturbations of the planets. — The solar system, as 
we have seen, consists of many bodies ; and each one of them 
attracts every other one, and attracts it more, according as it 
is nearer and more massive. Hence, no planet can continue 
to pursue the same elliptic orbit about the sun, as if the sun 
and planet were the only bodies. Nor can any satellite de- 
scribe its orbit undisturbed about the primary. The number 
and variety of these disturbing forces exerted within the system 
are very great. But many of them are so minute as to be in- 
sensible. As was shown in Chapter X. respecting the moon, 
so in regard to every planet and satellite, the disturbing influ- 
ences are of various kinds, some tending to alter the plane of 
the orbit, others to change its form, etc. These perturbations, 
like those of the moon, are classed into periodical and secular. 

338. Betrogradation of nodes. — If the orbit of a planet is 
oblique to that of another, one component of the disturbing 
force tends to move the nodes of the two orbits backward, as 



174 PERTURBATIONS OF THE PLANETS. 

shown in Art. 192. And every satellite, whose orbit is inclined 
to that of its primary, is acted on by the sun, in the same man- 
ner as the moon is : its nodes retrograde on the orbit of the 
primary. For the planets, this retrograde motion is excessively 
slow, generally amounting to only a few minutes in a century. 

339. Change of inclination. — Another disturbance of the 
orbit of a planet has respect to its inclination to the orbit of 
.another. There are small periodical oscillations in the inclina- 
tion, at every revolution, which nearly compensate each other, 
like those of the moon's orbit (Art. 193). But the compensa- 
tion not being exact, there is a minute change, which remains 
unbalanced, and accumulates for many centuries, when the 
change is reversed and accumulates in the opposite direction. 
These secular oscillations, however, are all within narrow 
limits. Thus, the ecliptic, though generally spoken of as a 
fixed plane, is not truly so, but is subject to a minute change 
of a few seconds in a century. It is proved that the whole 
variation can never amount to 3°, and that within that range 
it will occupy many thousands of years in making a single sec- 
ular oscillation. 

340. Advance of apsides. — All planets within the orbit oi 
a given planet, conspire, on the whole, to increase its gravity 
toward the sun ; while the general effect of those outside ot 
the same orbit is to diminish it. It was shown (Art. 183) that 
the sun, being outside of the moon's orbit about the earth, 
sometimes increases and sometimes diminishes the moon's ten- 
dency to the earth, but on the whole diminishes it. The same 
thing is true of every planet outside of the orbit of another. 
One consequence of the change in the attraction is, to cause 
the line of apsides to advance and to retrograde alternately ; 
but the resultant of the whole action is an advance. The 
earth's apsides advance 11-J" in a year (Art. 147). So the line 
of apsides of most of the planetary orbits has a slow motion 
from west to east. 

341. Change of eccentricity.— A planet tends to increase 
the eccentricity of an orbit within its own, when the two 



PEKTEEBATIONS OF THE PLANETS. 175 

planets are in its line of apsides at conjunction and opposition, 
and to diminish it when the line from the sun to the outer planet 
makes a right angle with the line of apsides; analogous to 
the action of the sun on the moon's orbit (Art. 187). These 
disturbances are very minute, but they will not balance each 
:>ther during a svnodical revolution ; and therefore there is a 
small secular change in the eccentricity of the orbits. For ex 
ample, the eccentricity of the earth's orbit has been diminish 
ing, and for many thousands of years to come it will continue to 
diminish, at the rate of 0.00004 per century. The orbit, how- 
ever, will never reach the exact form of a circle, but after 
arriving to a minimum of eccentricity, it w T ill begin to return 
to a more eccentric form, and thus will oscillate about a mean 
value perpetually. And the range of its eccentricity is so 
limited, that the ellipse, if correctly represented, can never 
diner visibly from a circle. 

It is this slow change in the earth's orbit which causes the 
secular inequality of the moon's motion (Art. 195). 

342. Change in the length of the major axis. — There are 
also minute periodic changes in the length of the major axis oi 
an orbit ; that is, in the mean distance of a planet from the 
sun. But both calculation and observation establish the fact, 
that there is no secular inequality, because the periodical 
changes exactly compensate each other. And, if the mean dis- 
tance of each planet from the sun has permanently the same 
value, then, according to Kepler's third law, the periodic time 
is also constant. * 

343. Long periods. — There are in the solar system several 
cases of inequality, accumulating for centuries, which never- 
theless have the character of periodical rather than secular 
inequalities, and depend on the fact that the periodic times oi 
two planets almost exactly measure a certain length of time. 

For example, the earth makes 8 revolutions in very nearly 
the same time in which Yenus makes 13. Hence, every fifth 
conjunction occurs within 1J° of the same points of their re- 
spective orbits. At the end of this period of S years, there is 
a minute perturbation, which remains uncompensated, and 



176 PERTUKBATIONS OF THE PLANETS. 

which is about doubled at the end of 16 years, and tripled at 
the end of 24 years, and so on. This disturbance is very small, 
never amounting to more than a few seconds ; but it requires a 
period of 240 years in order to pass through all its changes. 

The long inequality of Jupiter and Saturn is a more remark- 
able case. Jupiter makes 5 revolutions, and Saturn 2, in 
nearly the same time. An unbalanced disturbance, which ap- 
pears at the end of this time, goes on accumulating. During 
the 17th century, Saturn was constantly retarded, and Jupiter 
accelerated. But in the 18th, this was reversed, and Saturn is 
now accelerated, and Jupiter retarded. This will continue still 
longer ; and the whole period required for this inequality is 
more than 900 years. The deviation, at its maximum, is 49' 
for Saturn and 21' for Jupiter. 

344. Degree of change in the several elements. — Of the sev- 
eral elements named at the beginning of this chapter, we see, 
from what precedes, that the following classification may be 
made. 

1. The 1st and 2d have no secular inequality whatever. 
Their value remains constant from age to age. The perma- 
nency of these two elements secures a constant length of the 
year, and a constant amount of heat from the sun on each 
planet. 

2. The 3d and 6th elements have small periodical oscilla- 
tions, but their secular change is in one direction ; the nodes 
perpetually retrograde, the apsides perpetually advance. But 
the continual change in the same direction in these two ele- 
ments has no tendency to derange the condition of things on a 
planet. As to the well-being of the occupants of a planet, it is 
of no consequence how the major axis of its orbit is situated, 
if only the form of the ellipse is preserved. It is also im- 
material in what direction the line of its nodes may happen 
to lie. 

3. The 4th and 5th elements have both periodical and secu- 
lar inequalities; but they range within very nanow limits, 
The smallness of these changes insures all the planets against 
any considerable change from year to year, in respect to the 
extremes of heat and cold, and in respect to the seasons. 



STABILITY OF THE SYSTEM. 177 

34o. Stability of the system. — Several of the secular in- 
equalities, before their true character had been demonstrated, 
excited great interest among astronomers, because they seemed 
to indicate the ultimate derangement of the order and stability 
of the system. If the eccentricity of the earth's orbit should 
continue to change in the same direction perpetually, the earth 
would at length, though perhaps not in millions of years, be- 
come unfit to be the habitation of man, because of the terrible 
extremes of heat and cold at perihelion and aphelion. So, ii 
the inclination of equator and ecliptic should continue its 
change perpetually in the same direction as at present, the sea- 
sons would by and by disappear, and afterward run to an ex- 
treme which would produce desolation over the whole surface 
of the earth. And if the secular inequality of the moon's 
period were always to go on as it has done for centuries past, 
the moon would at length be precipitated on the earth. 

But La Grange, La Place, and others have demonstrated 
that all the perturbations have their limits, and, after in- 
creasing with extreme slowness for many ages, must again de- 
crease in like manner ; and, furthermore, that the entire series 
of changes lies within so narrow bounds, that eo disastrous 
consequences can ensue. 

346. Manner in which the stability is secured '. — The sta- 
bility of the system is secured by the fulfilment of certain 
essential conditions in the arrangement of its parts. 

1. The great mass of matter constituting the solar system is 
in the central body, the sun being 700 times as great as all the 
other bodies united. Hence, all movements are principally 
controlled by the sun. 

2. The planets, and especially the large ones, are at great 
distances from each other; and thus the sun's influence over 
each is but little modified by their mutual attractions. 

3. The orbits, especially of the largest planets, have but 
slight eccentricity, and, therefore, always maintain their great 
distances from each other. 

4. The mutual inclinations of the orbits are small. Hence, 
there are no large forces operating to change the position of 
orbits, and thus disturb the seasons. 

12 



liO RELATIONS OF ELEMENTS. 

347. ^Relations of the planetary motions. — The planets arc 
so adjusted to each other, in respect to their velocities, dis- 
tances from the sun, periodic times, and gravity toward the 
sun, that if any one of these relations between two planets is 
known, all the others become known also. 

Let r — mean distance ; t = periodic time ; v = velocity 

g = gravity. Also, let s (slowness) = — , the reciprocal of ve 
locity ; and I (lightness) = — , the reciprocal of gravity. 

if 

v = 2 -f oo L (Art. 92) ; 

.*. v 2 oo -5-. But by Kepler's third law, 
t 

r 2 1 

f oo r s : .'. v 2 oo — , or v 2 oo — ■. 
r r 

A 1 1 A '« 1 l X A . 

As s = — , .*. v = — , and sr = -=- ; .'. -= oo — , and s oo r. 
v' s ' s 2 ' s 2 7 1 ' 

r r 3 £ a 

Again, since voo — , v z oo — . But, r 3 oo f ; ,\ v* oo — , 

to t 

or f x -;.'.- oo — , and s oo #. 
£ s t 

By the law of gravity, g oo — ; .\ — oo — - ; and Z oo r 2 . 

7' b T 

But s 2 oo ?' ; .'. s 4 oo r 2 , and s 4 go I. 

■ Bringing together these results, we find four variations, 

s oo s 1 ; r oo s 2 ; £ oo s 3 ; I oo <s- 4 . 

Hence, we have the reciprocal of velocity, s / the distance, 
r j the periodic time, t / and the reciprocal of gravity, I / re- 
spectively denoted in their ratios by the geometrical series, s\ 
s 2 , s s , s\ in which the first term and the ratio are equal. 

348. Mode of using these variations for calculation. — If 
the velocities of two planets are given, we first take their re- 
ciprocals, and thus have the ratio of s for the two. The terms 
of this ratio are then raised to the second, third, or fourth 
power, according as we wish to compare r, or t, or L 



RELATIONS OF ELEMENTS. 179 

But if the ratio of distances, or times, or gravities is given, 
the corresponding root is first extracted, in order to find the 
ratio in respect to s, and then we proceed as before. 

349. Examples. — 

1 . The planetoid Pallas has a period of 4| years ; how much 
further is it from the sun than the earth is ? How much less is 
it attracted % How much slower does it move ? 

Let t, «, ?', I be used for the earth, and T, S, R, L for Pallas 

Then, t : T : : 1 : 4.667; 

,\ 1* : (4.667)* : : s : S ; 
.*.«§: S :: 1 : 1.67 ; that is, the earth's ve- 
locity is 1.67 times as great as that of Pallas. 

Again, r : R : : l 2 : (1.67) 2 : : 1 : 2.7926 ; or, Pallas is 2.7926 
times as far from the sun as the earth is. 

Again, I : L : : l 4 : (1.67)* : : 1 : 7.7985 ; therefore, the earth is 
attracted by the sun about 7.8 times as much as Pallas is. 

2. What would be the period of a satellite revolving about 
the earth close to its surface ? 

The distance of this satellite to that of the moon is as 1 : 60 ; 

\ s : 8 : : 1 : (60)*; .*. t : T : : 1 : (60)* : : 1 : 464.66. 

But the moon's period is 27.32 days, or 655.68 hours. 
Hence, the period of the satellite is 1.411 h., or lh. 24m. 39s. 
nearly. 

3. How much faster must the earth rotate on its axis, in 
order that bodies on the equator may lose all their weight % 

This is just the condition of the body in example 2d, whose 
period is 1.411h. But the earth's time of rotation is 24h., 
which is 17 times 1.411h. Therefore, if the earth were to 
rotate 17 times more rapidly than at present, all bodies on 
the equator would just lose their weight, and revolve inde- 
pendently. 

4. What would be the periodic time of a body revolving 
about the earth, at the distance of 5,000 miles from the 
center? Ans. lh. 59m. 23JS. 



180 COMETS. 

5. What must be the moon's distance from the earth, in 
order to revolve about it once in a year ? 

Arts. 1,344,000 miles. 

6. Suppose a planet to be discovered, whose daily velocity is 
5 times as great as that of Mercury, what is its distance from 
♦he sun's center? Ans. 1,430,000. 



CHAPTEE XVIII. 

COMETS.— SHOOTING STARS. 



350. A comet defined. — A - comet is a body which consists 
of nebulous matter, and revolves about the sun in a very eccen- 
tric orbit. Most comets present a roundish ill-defined appear- 
ance, often having a bright central part called the nucleus. 
The fainter part, surrounding the nucleus, is called the coma 
{hair) ; and the tail, which distinguishes many comets, is 
merely the extension of the coma. It is the streaming ap- 
pearance of the tail, resembling hair, which gave the name 
u comet" to this class of bodies. The nucleus has been some- 
times supposed to be solid ; but it probably consists always of 
nebulous matter in a more condensed state than the other 
parts. The nucleus and coma are called the head of the 
-comet. 

351. Number of comets. — Many hundreds of comets have 
Jbeen recorded, most of them, of course, visible to the naked 
eye. But lately it is observed that most comets are telescopic 
objects. And many which would otherwise be seen, escape 
observation by being above the horizon only in the daytime. 
The whole number, therefore, belonging to the solar system 
is undoubtedly to be reckoned by thousands, or tens of thou- 
sands. 

352. Eccentricity of orbit. — All known cometary orbits are 
*nore eccentric than any planetary orbit ; and most of them are 



ECCENTRIC ORBITS. 181 

exceedingly so, their perihelion being as near the sun as Mer- 
cury and Yen us, or nearer, and their aphelion as far off as the 
most distant planets, or even beyond. And some appear to be 
^ellipses of infinite length — that is, parabolas ; while others ex- 
hibit the form of hyperbolas. In orbits of these last forms, 
comets can, of course, pass the perihelion but once. 

353. Consequences of great eccentricity. — 

1. One effect of this great eccentricity is, that a comet is too 
far from the earth to be seen, except during a small part of its 
revolution, while it is near the center of the system. 

2. Another effect is, that great changes take place in the 
condition of the nebulous matter of which the comet is com- 
posed. As a comet approaches the sun, both the nucleus and 
coma grow less in diameter, and enlarge again as it departs 
But the tail, if there is one, is rapidly lengthened as the comet 
approaches, and is diminished in length when it withdraws. 
Sometimes, a comet, whose appearance is spherical when first 
seen, begins suddenly to exhibit the formation of a tail as it 
comes nearer, which at length stretches over a large arc of the 
-sky; and after the perihelion passage, as it departs from the 
sun, the tail wholly disappears before the comet becomes in- 
visible. 

It might be supposed that this diminution of the coma re- 
sults from the loss of material which is taken away to form the 
tail, while the comet is approaching the sun ; and that the sub- 
sequent enlargement is due to the return of the same material, 
as the tail is contracted. But this will not fully explain the 
observed changes; for the contraction and subsequent expan- 
sion occur when no tail is formed. Hence, it is supposed that 
the heat of the sun reduces the dimensions of the nucleus by 
expanding a portion of it into the coma, and also changes the 
nebulous matter of the coma into a pure, transparent gas, which 
is afterward condensed into a visible form again, as the comet 
withdraws from the sun. 

354. Form and direction of tails of comets. — The forms oi 
tails belonging to different comets are exceedingly varied. In 
general, however, the sides diverge from the head, so that the 



182 



TAILS OF COMETS. 



most distant and faintest part is broadest, as in the comets ot 
1680 and 1811 (Figs. 82, 83). In some cases, the divergency is 
very slight, as in the comet of 1843 (Plate I., at the end). 



Fig. 82. 



Fig. 83 





COMET OF 1811. 



COMET OP 1680. 



Not nnfrequently, the principal light of the tail appears to< 
proceed from its edges, presenting somewhat the aspect of two 
tails diverging from the sides of the coma. In such cases, the 
coma and tail seem to have the form of a hollow paraboloid, 
so that we look through a much greater extent of illuminated 
matter on the sides than in the central parts. In the comet of 
1858, the nucleus was at one time surrounded by a series of 
parabolical envelopes, which increased in number as the comet 
approached the sun. (PL II., Fig. 1). 

In a few instances, the tail has been known to consist of sev- 
eral luminous rays, diverging from each other, as the comet of 
1744, in which there were six, the extreme ones making with 
each other an angle of about 45°. 



DIMENSIONS OF COMETS. 183 

The general direction of the tail is from the sun; so that, as 
a comet approaches the sun, the tail follows it ; but as it re 
cedes, the tail is directed forward. The axis of the tail is not, 
however, a straight line, but more or less curved backward, so 
that the convex side of the curve is foremost in the motion. 

355. Cause of the direction and curvature of the tail. — • 
Modern telescopic observations on some of the most conspicu- 
ous comets, show that the material of which the tail is formed 
is first projected toward the sun, rather than from it ; and that 
some force emanating from the sun then drives it, with great 
velocity, in the opposite direction, causing it to sweep past the 
nucleus on both sides, and stretch millions of miles into space. 
The rate at which it is thus driven from the sun is sometimes 
enormous. In the case of Halley's comet, in 1835, the nebu- 
lous matter had a velocity of 2,000,000 miles per day. In Do- 
natio (1S5S), it reached the rate of 8,000,000 miles per day. 
What force it is which the sun thus exerts in a direction oppo- 
site to its gravity, it is vain to conjecture. It must be supposed 
that at least a part of the material driven so violently from the 
comets is dissipated and lost ; and there is indication of this in 
the diminished size and brilliancy of those whose returns have 
been noticed. Perhaps the numerous comets which have no 
tails have been divested of them by this process. 

The bending of the tail backward is a necessary consequence 
of the longer arc which the extreme part of the tail must de- 
scribe. The material of the tail has the same velocity in the 
orbit as the head, when it is driven from it. This velocity it 
retains ; but, having to describe a curve about the sun several 
millions of miles outside of the other, it must, of course, fall 
behind it. 

356. Dimensions of comets. — The dimensions of comets are 
various, and, on account of their nebulous character, they 
never admit of accurate measurement. The nucleus of a large 
comet is sometimes 5,000 miles, and the coma 200,000 miles 
in diameter, wiiile the tail has, in one case, attained the extra- 
ordinary length of 200,000,000 miles. 

The apparent length of a comet's tail is often sufficient to 



184 DIRECTIONS OF COMETARY MOTIONS. 

span an arc of 20° or 30° on the sky, and sometimes mucls 
more than this. The comet of 1680 extended 97°, and that o^ 
186 1 , 106°. The fainter part, in all cases, is seen only by in- 
direct vision. 

It is obvious that the real length can not be inferred from the 
apparent, nntil the distance from us, and the obliquity to 
our line of vision, are obtained. 

357. Light of the comets. — These bodies, like the planets 
and satellites, shine by solar light which they reflect to us„ 
But, unlike all planetary bodies, they are in a condition so- 
attenuated, that the sun's rays penetrate every part of them 
without obstruction. The brightness of a star is not diminished 
in the least when seen through the tail or coma of a comet. In. 
a few instances, a star has been seen through the nucleus, and 
even then was not essentially dimmed. 

A satisfactory proof that the comets are seen by the sun's 
light which they reflect, is, that their brightness diminishes as- 
they recede from the sun ; so that they are at length lost to 
view, not by being too small to fill an appreciable angle, but 
too faint to be visible. This would not be true of a self-lumi- 
nous body : its brightness would remain the same at all dis- 
tances from us ; that is, its light would diminish no faster 
than its apparent area. Appendix G. 

358. Quantity of matter in comets. — Though some of the 
largest comets surpass all other bodies in the solar system in 
magnitude, yet in respect to their mass they are too small to 
have produced as yet the slightest perceptible effect. They 
sometimes come very near planets and their satellites, but are 
never known to exert the least influence on them. They do,., 
of course, attract the planets, because they are attracted by 
them, and suifer great disturbances from them. But until 
they themselves produce some effect which is appreciable, their 
mass must be regarded as infinitely small. 

359. Directions of cometary motions. — The cometary orbits 
are unlike the planetary, not only in the degree of their eccen- 
tricity, but in the varied positions of their planes. Instead of 



ORBITS OF COMETS. 185 

>eing limited to a narrow zone like the zodiac, they make 
3very variety of angle with the ecliptic, so that a comet is as 
likely to pass round the sun from north to south as from west 
to east. And whether the orbit is much or little inclined, the 
comet's motion in it is as often retrograde as direct. 

360. Means of determining a corners orbit. — Since a comet 
can be seen only during the time of its describing a short arc 
near the perihelion, the astronomer has not the same oppor- 
tunity for fixing its orbit as he has in the case of a planet, 
which can be observed in all parts of its course. 

It is true in theory, that by any three observations on the 
position of a body revolving about the sun, its whole orbit can 
be determined. But if it is very eccentric, and the observa- 
tions are confined to a small portion at one extremity, the 
slightest error may greatly change the distance of the aphelion, 
and consequently the length of the axis and the periodic time. 

It is usual, therefore, to assume the path to be a parabola — 
that is, an ellipse of infinite length, whose eccentricity is 1. 
There are then but four of the seven elements (Art. 325) to be 
determined — namely, the 3d, 4th, 6th, and 7th. But instead of 
the 2d element in Art. 325, there may be substituted the peri- 
nelion distance, making in all five elements, as follows : 

2. The perihelion distance. 

3. The longitude of the ascending node. 

4. The inclination of its orbit to the ecliptic. 

6. The longitude of the perihelion. 

7. The place of the body at a given time. 

These five elements may usually be determined without 
much difficulty. 

361. Process of finding the five elements. — The right 
ascension and declination of the comet are observed on every 
favorable night with all possible care, and the exact time of 
every observation is recorded. Three of these dates are selected, 
several days apart ; and from the right ascension and declina- 
tion at each date are deduced the geocentric longitude and 
latitude of the comet. The heliocentric places of the earth 
at the same times are known, since each is 180° from the 



186 COMETARY ELEMENTS. 

sun's apparent place at the same time. If we imagine threti 
straight lines to be drawn from the known places of the earth 
through the corresponding positions of the comet, its distance 
from ns in each line must be determined by the following con 
ditions, in accordance with Kepler's laws : 

1. A plane passing through the three positions must also 
pass through the sun. 

2. The three places must be in a parabola, whose focus is at 
the sun. 

3. The areas included between the radii vectores drawn to 
the sun must be proportional to the times. 

Points are successively assumed in the given lines, until at 
length those are found which will fulfill the above conditions. 
By this tentative process the orbit is approximately deter- 
mined. 

Three other dates may then be tried in the same manner ; 
and if the results nearly agree, the mean may be considered 
more accurate than either. 

This method is just as applicable for determining, approx- 
imately, the orbit of a newly discovered planet or planetoid. 
But in these cases, the observations can usually be followed 
up in various parts of the orbit, and thus previous errors cor- 
rected. 

362. Determination of the remaining elements. — If a comet 
remains in sight for several months, it is quite probable that 
long-continued and careful observations will show that the orbit 
is not truly a parabola, but an ellipse. In such a case, the 1st 
and 2d elements (Art. 325) may be computed, and the 5th 
corrected. Though there may. be good evidence that the orbit 
is an ellipse, yet there must be great uncertainty in any deter- 
mination of periodic time, mean distance, and eccentricity, until 
they are settled conclusively by a return of the comet to its 
perihelion. 

A comet, on its return to its perihelion, is not to be identified 
so much by its physical aspect, as by the agreement of its ele- 
ments. If the place of the node, the inclination to the ecliptic, 
the place of the perihelion, and the perihelion distance agref 
very nearly with those of some comet, which has been prev: 



HALLE F'S COMET. 187 

ously seen, it is fairly presumed to be the same, even though its 
appearance may essentially differ. 

When a comet is thus identified, its periodic time is, 01 
course, known ; and from this, its mean distance, and the ec- 
centricity of its orbit are readily obtained. 

363. Comets whose elements have been computed, but not 
verified. — The orbits of more than 300 comets have been com- 
puted. But a great majority of these appeared to be parabolas, 
and no prediction of their return could be made. In about 60 
cases, the movement of the comet seemed to afford evidence 
that the orbit was an ellipse, and in 7 others, a hyperbola. 
For the elliptic orbits, the returns were, of course, predicted. 
But most of the computed periods are long, generally hundreds, 
and, in several instances, thousands of years ; so that as yet 
very few of them have been verified. The period of a comet 
seen in 1849 was calculated to be 2,115 years. If the compu- 
tation be supposed correct, the distance of this comet from the 
sun at aphelion is about 11 times the distance of Neptune, and 
its next return will occur in the year 3964. But the results of 
calculation, in such cases, are exceedingly uncertain. The cir- 
cle NE (Fig. 84) represents the orbit of Neptune ; S, the sun ; 
ana SC, the orbit of the comet of 1849. The focus is very close 
to the vertex of the ellipse, as represented by the dot 

Fig. 84. 



f <S ^SZZZ 



364. Halley's comet — This is the only comet of long period, 
the elements of whose orbit are all known. It is a comet oi 
considerable splendor, and describes its orbit in 75 or 76 years. 
Halley observed it in 1682; and finding, by computation, that 
its path was nearly identical with those of the comets of 1607 
and 1531, he conjectured that the three were one and the same 
comet, and predicted that i A would return early in 1759. It 
did return March 12th of that year, and again on the 16 th of 



188 COMETS OF SHOET PERIODS. 

November, 1835. On its last return, it reached the perihelion 
within two days of the calculated time. What renders such 
agreement remarkable is not merely the great length of the 
periodic time, but the great allowance to be made for the dis- 
turbing influence of the principal planets. On its last return 
but one, the period of this comet had been increased nearly 
two years by the attractions of Jupiter and Saturn. The aphe- 
lion is nearly 600,000,000 miles beyond the orbit of Neptune. 
Figure 85 presents the form of the orbit of Halley's comet, and 
its magnitude in comparison with the larger planetary orbits. 
The eccentricity is nearly 0.97 ; hence the distance of the focus 
from the vertex is only .03 of the semi-major axis. The dot at 
the left hand correctly represents the place of the focus occu- 
pied by the sun. 

Fig. 85. 




365. Comets of short period, whose orbits are known. — 
The eight comets in the following table are known by the 
names of the persons who either discovered them, or first pre- 
dicted their returns. 



COMET. 


Period 
in years. 


Perihelion Distance. 


Aphelion Distance. 


Encke's 


3| 

H 

H 

H 

7i 

13| 


32,000,000 

110,000,000 

70,000,000 

64,000,000 

82,000,000 

108,000,000 

192,000,000 


387,000,000 
475,000,000 
510,000,000 
537,000,000 
585,000,000 
530,000,000 
603,000,000 


De Yico's 


Winnecke's 


Brorsen's 


Biela's 


D'Arrest's 


Fave's 


Mechain's 





biela's comet. 189 

It will be perceived by the table, that these comets consider- 
ably resemble each other in period and distance. They are 
also alike in being telescopic, and nearly or entirely destitute 
of tails, and in moving from west to east, excepting the last, at 
inclinations to the ecliptic not larger than 15° or 20°. 

366. A resisting medium. — Two of the above comets, 
Encke's and Faye's, have given decided indications of acceler- 
ation in their orbits. This shows that they meet with some 
obstruction, which diminishes their projectile force, in conse- 
quence of which the centripetal force draws them into a smaller 
orbit, which is, of course, described in less time. 

Some suggest that as the received theory of light requires the 
existence of a medium throughout space, a substance of so little 
density as a comet may possibly be obstructed by it sufficiently 
to render the diminution of period perceptible. 

According to others, the obstruction may arise from collision 
with innumerable small bodies which revolve about the sun. 
The earth is meeting with such bodies incessantly, as is proved 
by the numerous shooting stars which are continually striking 
into the atmosphere. It is reasonable to suppose that other 
bodies, as well as the earth, also meet them ; and that the 
thinnest and lightest bodies, such as the comets, should show 
the effect of such collisions. 

If Encke's and Faye's comets, in either of these -ways, are 
gradually diminishing their periodic times, then every other 
comet must by and by exhibit the same change ; and the time 
will come eventually when all this class of bodies will, at their 
respective perihelia, approach so near as to fall upon the sun, 
and be combined with its substance. 

367. Division of Biela's comet. — One of the most remark- 
able facts which has occurred in the history of comets was the 
division of Biela's comet into two distinct comets. This ap- 
pearance was first noticed at its return, in 1846. The two 
comets were unequal in size, and the larger had a short tail ; 
the other was only a little elongated ; whereas, before the di- 
vision, the comet was spherical, without any appearance of & 
tail. Since the division took place, the two bodies have moved 



190 BEMAKKA.BLE COMETS. 

m separate and independent orbits. Their distance apart in 
1852 was about 1J millions of miles. Appendix H. 

368. Other remarhMe comets. — A few other comets are 
here mentioned, which, on account of their splendor, or foi 
some other reason, are regarded as objects of special interest. 

369. The comet of 1680.— (Fig. S3). This was a comet ol 
unusual brilliancy, and appears to have been the first whose 
elements were calculated by Newton. At perihelion, its center 
was only 130.000 miles from the surface of the sun; so that, if 
its diameter was as large as that of many comets, it must have 
come in contact with it. Its velocity at perihelion was suffi- 
cient to have carried it round the sun at that distance in less 
than three hours. 

370. The comet of 1744. — This was the most splendid 
comet of the 18th century. The remarkable features of it 
were, its great brightness, and the number of its tails. Its 
light was nearly equal to that of Yenus, and it was distinctly 
seen in the daytime, even by the naked eye. After passing 
the perihelion, its tail was spread into six distinct branches, 
near 40° in length, and the extreme ones diverging about as 
many degrees from each other. 

371. The comet of 1770. — The great interest which at 
taches to this comet arises from the fact that it has twice suf- 
fered a great change of orbit, in consequence of the disturbing 
action of Jupiter. It first appeared in 1770, shining with con- 
siderable splendor. In 1776, it again passed the perihelion, 
and has never been seen since. Computations made by La 
Place and others showed that, before its first appearance, it had 
revolved in a large orbit, beyond our vision, and had a period 
of 48 years. In 1767, it came near Jupiter, and lost so much 
of its velocity, that it was drawn into a small orbit, whose 
perihelion was far within the orbit of the earth. As two of its 
revolutions were about equal to one of Jupiter, it was predicted 
that it would be again subjected to a great disturbance at its 
aphelion. This actually took place in 1779, so that it has 



REMARKABLE COMETS. 



191 



never returned to our view. Its present period is calculated to 
be about 20 years. AB (Fig. 86) is a part of Jupiter's orbit ; 
E is the earth's orbit ; CD is the path 
of the comet before 1767. Near D. 
it was so retarded by Jupiter, that the 
sun drew it into the small orbit DFH, B 
which it described twice, when, uear 
D, it was again powerfully affected 
by Jupiter, and received a great accel- 
eration, which caused it to pass out 
once more into a large orbit, DK. 

37 2. The comet of 1843.— The 
brightness of tiiis comet was so great, 
that it was seen during the day. 
Its perihelion distance was less than 
550,000 miles, and its exterior parts 
were probably in actual contact with 
the sun. ~No other comet has been 
known to approach so near. The tail spanned 70° of the 
sky, and was unusually straight and slender, as exhibited in 
PL I. 




373. The comet of 1858. — This is also called Donati's 
comet, having been discovered by Donati, of Florence. It was 
remarkable for the series of envelopes formed successively 
about the -nucleus, as it approached the perihelion. The ap- 
pearance of the head is shown in PL II., Fig. 1, and the entire 
comet in Fig. 2. Its period is computed to be about two 
thousand years. 

374. The comet of 1861. — This comet came so near the 
earth, that it is believed a part of the tail swept across it. But 
it is not certain that any visible effect was produced. The 
apparent length of its tail, at one time, was 106°. Fig. 87 
shows its form at that time. 



37 5. Effects of collision between a planet and a comet. — 
Whether a direct collision between the earth and the nucleus 



192 



SHOOTING STARS. 



Fig. 87. 



! J f ; V%f Ijlfft 

•IS:' 







of a coinet would produce serious ef- 
fects, it is impossible to know, because i 
so little is understood respecting the 
density of the nucleus. But the coma 
and tail consist of matter thousands of I 
times more rarefied than the earth's 
atmosphere, and would probably fail I 
to penetrate it at all. The earth is 
thought to have passed through a 
comet's tail, at least in one instance, 
but without producing any perceptible 
effect. 

376. Shooting stars. — This is the 
popular name given to those bodies 
which, appear like stars or planets mov- 
ing across some part of the sky, and 
then vanishing. They are equally well 
known by the name of meteors. They 
may be seen in any clear night, by 
watching an hour or two, especially if 
the moon is not shining. 

377. Height a?id velocity. — By means 
of concerted observations, made at sta- 
tions quite distant from each other, the 
angle can be measured, which is in- 
cluded by lines drawn from a meteor to the stations, both at 
the beginning and end of its motion, and thus its distance 
and velocity can be measured. The heights of meteors are 
thus found to be generally about 50 miles, and their velocities 
20 or 30 miles per second. Coming into the air with such 
great velocity, they are almost instantly set on fire, and their 
substance becomes incorporated with the atmosphere. 




378. Gaseous meteors. — If the ordinary meteors were more 
dense than a gas, they would hardly lose all their motion, as 
they do, before reaching the earth. The most interesting facts 
relating to this class of bodies are the following : 



SHOOTING STARS. 193 

1. They often occur in showers — that is, thousands and hun- 
dreds of thousands of them are seen in a single night. 

2. These showers have periodical returns. 

3. The meteors of a shower come into the atmosphere in a 
given direction, or, in other words, in parallel lines. The op- 
tical effect is, that they appear to describe arcs of great circles, 
having a common place of intersection. 

379. Dates of meteoric showers. — The most remarkable 
meteoric shower of the present century was November 12-13, 
1833. Not less than 200,000 meteors were seen during the 
night at any one station. Like showers occurred at the same 
time in 1799 and 1866. And generally, there are more meteors 
about the 12th of November than at any other time of the year. 

Other dates at which meteors are unusually abundant are 
April 21st, August 10th, and December 7th. 

380. Origin of the gaseous meteors. — The known motion of 
the earth, and the observed velocity and direction of this class 
of bodies, lead to a knowledge of their heliocentric motions. 
It is found in this way that they describe ellipses about the 
sun, and are therefore to be regarded as minute cometary 
bodies. Those which come in showers seem to belong to ex- 
tensive groups, which revolve about the sun in zones or rings. 
There appear to be three or four of these zones, whose planes 
are situated at different obliquities to the ecliptic, and across 
which the earth passes once a year. When the earth traverses 
a more crowded portion of such a ring of meteors, the phenom- 
enon of a meteoric shower occurs. Appendix I. 

381. Solid meteors. — There is another class of meteoric 
bodies, which afford indubitable evidence of being solid. Like 
the gaseous meteors, they plunge into the atmosphere with 
great velocity, and are inflamed by the violent attrition. Be- 
fore reaching the earth they usually explode, and scatter their 
fragments. Some of them, however, appear to lose only small 
portions of their mass by explosion, and pass on in their orbits 
around the sun — greatly disturbed, of course, by the earth's at- 
traction. 

13 



194: THE STELLAR UNI TERSE. 

382. Aerolites. — This is the name usually given to the frag 
ments thrown down by solid meteors ; though in rare instances, 
an aerolite obviously constitutes the entire meteor itself. Aer- 
olites consist of iron, silex, and a few other materials, which 
are all known among terrestrial substances. But they are 
always distinguishable from terrestrial bodies by their peculiar 
structure. Since the great velocities of meteors, solid as well 
as gaseous, have become known, the former theories as to the 
origin of meteoric stones, or aerolites, have been abandoned. 
Such velocities, if they could be generated at all on the earth, 
could never exist in horizontal or downward directions. Both 
solid and gaseous meteors are therefore considered as describ- 
ing orbits about the sun. The interplanetary spaces, which 
have been generally reckoned as vacant, may perhaps be to a 
great extent occupied by innumerable bodies, of a grade far 
below that of comets and planetoids. 



CHAPTEK XIX. 



THE FIXED STARS. — THEIR CLASSIFICATIONS. — THEIR DIS- 
TANCES AND MOTIONS. — DOUBLE STARS, CLUSTERS, AND 
NEBULiE. — THE NEBULAR HYPOTHESIS. 

383. The stellar universe. — The bodies described in the 
foregoing chapters all belong to the solar system. If our inves- 
tigations are extended outside of this system, we find that there 
are other systems, greater or less than this, unlimited in num- 
ber, and separated from the solar system and from each other 
by solitudes so vast, that each system is only a point in com- 
parison with the distances between them. The central sun in 
each of these countless systems is a fixed star. 

The word " universe" is employed to express the sum total 
of all these systems, the number of which, and the extent of 
space occupied by them, are utterly beyond the reach of human 
comprehension. 

384 The fixed stars* and their magnitudes. — The fixed 



MAGNITUDES OF STARS. 10& 

stars are so called, because, to common observation, they always 
maintain the same situations with respect to each other. AH 
the thousands of bright points ordinarily seen in the sky by 
night are fixed stars, with the exception of two or three, possi- 
bly four, which are planets. 

The fixed stars are classified according to magnitudes, thongb 
the word, when thus used, signifies only degrees of bright?} ess. 
The stars which can be seen by the naked eye, in the most 
favorable circumstances, are divided into six magnitudes. 
Those which can be seen only by the aid of the telescope, 
called telescopic stars, are arranged into several more ; so that 
all the magnitudes are 16 or 18. 

Stars of the same magnitude are not equally bright ; for 
there is a continual gradation in respect to brightness ; so that, 
if the intensity were accurately measured, probably the light 
of but very few would be found exactly equal. 

Stars of the first magnitude are fewest in number, and, gen- 
erally, the smaller the magnitude, the larger the number of 
stars included under it. The limits of the successive magni- 
tudes differ somewhat, according to different astronomers ; but 
the following round numbers do not vary widely from any of 
them. 



1st magnitude . 


. 20 


4th magnitude . 


. 300 


2d " 


. 40 


5th " 


. 950 


3d « 


. 140 


6th " 


. 4450 



*n all, near 6,000, visible to the naked eye. The numbers of 
the telescopic stars increase at so rapid a rate, that they have 
to be reckoned by millions. 

385. Cause of unequal brightness. — We might suppose 
either that the stars are themselves unequal in respect to the 
quantity of light which they emit, or that they appear un- 
equally bright on account of their different distances. It is 
undoubtedly true that there is some diversity in the bodies 
themselves ; and yet, the rapid increase of numbers as the mag- 
nitudes are less, indicates that difference of distance is the chief 
cause of inequality in brightness. If there is any approach to 



196 



CONSTELLATIONS. 



a uniform distribution of the stars in space, those which are 
nearest should be fewest in number, and should, in general 
appear brightest. 

386. Constellations. — The fixed stars are also classed topo- 
graphically in constellations. This division is very ancient; 
and some of the constellations are mentioned by the earliest 
writers. The names given to them are those of the animals, 
heroes, and other objects of pagan mythology. 

Constellations of the zodiac. 



Aries. 


Libra. 




Taurus. 
Gemini 


Scorpio. 
Sagittarius. 




Cancer. 


Capricornus 




Leo. 
Yirgo. 


Aquarius. 
Pisces. 




Constellations north of the zodiac. 


Ursa Major. 
Ursa Minor. 
Draco. 


Auriga. 
Leo Minor. 
Canes Yenatici. 


Cygnus. 

Yulpecula. 

Aquila. 


Cepheus. 


Coma Berenices. 


Antinous. 


Cassiopeia. 
Camelopardalus. 
Andromeda. 
Perseus. 


Bootes. 

Corona Borealis. 

Hercules. 

Lyra. 


Delphinus. 

Pegasus. 

Ophiuchus, 


Constellai 


'ions south of the zodiac. 


Cetus. 


Monoceros. 


Hydra. 


Orion. 


Canis Major. 


Crater. 


Lepus. 


Canis Minor. 


Corvus. 


Centaurus. 


Crux. 


Eridanus. 


Lupus. 


Argo Navis. 





The foregoing are the principal constellations ; but several 
more, mostly small ones, may be found on globes and charts. 



ANNUAL PARALLAX. 191 

Within each constellation, the brightest stars are designated 
by the letters of the Greek alphabet in the order of brightness. 
Tims, a Lyrse, is the brightest star in Lyra ; /3 Scorpionis, the 
brightest but one in Scorpio, etc. After the Greek letters are 
all used, Roman letters, and then numerals, are employed. In 
some cases, the order of brightness does not accord with the 
order of the alphabet. This may result from a change of 
brightness, which has taken place since the stars were first 
named. When a capital letter follows a number, there is ref- 
erence to the catalogue of some astronomer. Thus, 84H is the 
star 84 of a certain constellation in Herschel's catalogue. 

A few conspicuous stars are still known by individual names 
given to them in ancient times ; as Arctnrus, Antares, Sirius, 
Yega, etc. 

The first catalogue of stars was made by Hipparchus, before 
the time of Christ, and contained 1,022 of the most conspicuous 
stars. Catalogues of the present day contain hundreds of thou- 
sands of stars, whose right ascensions and declinations are given 
for a certain date. 



387. Effect of telescopic power on fixed stars. — One indica- 
tion of the vast distance of the fixed stars is, that no power 
of a telescope sensibly magnifies them. Even under a power 
which increases the diameter of a body 5,000 times, they appear 
no larger than to the naked eye. It is inferred that they fill an 
angle so small, that 5,000 times that angle is still too minute to 
be perceived. Any appearance of disk which a star presents, 
either with a telescope or without, is the effect of the light 
upon the retina of the eye. It is called a spurious disk, since 
an increase of magnifying power causes no increase of its di- 
ameter. 



388. Annual parallax. — Another proof that the fixed stars 
•are at an immense distance from us, is the fact that while we 
shift our position every six months from one side of the earth's 
orbit to the opposite, a distance of 185,000,000 miles, there is 
no perceptible change in the relation of the stars to each other. 
It is only after long-continued and most accurate observation, 



198 ANNUAL PAKALLAX. 

that a few stars have been discovered to suffer an annual 
change of position, which is clearly of the nature of paral- 
lax. 

The annual parallax of a star is the angle, at the star, sub- 
tended by the radius of the earth's orbit. As this angle is in 
almost all cases too small to be detected, it shows that the 
earth's orbit, seen from the distance of the stars, appears as a 
mere point. 

389. The parallactic path of a star. — If the annual paral- 
lax of a star is in any case perceptible, its apparent movement 
during the year depends entirely on its situation in relation to< 
the ecliptic. 

A star in the plane of the ecliptic will appear to oscillate 
back and forth in a straight line once in a year. It will appeal 
stationary at the two opposite seasons, when the earth is going 
toward it, and from it ; and if we imagine a diameter of the 
earth's orbit joining these two positions, the star will seem to 
describe a straight line parallel to that diameter, its motion 
during each half-year being opposite to the general direction of 
the earth's motion. 

But if a star at the pole of the ecliptic should exhibit any 
parallax, its apparent motion would be in an orbit parallel to 
the earth's orbit, and similar to it : it may be regarded, there- 
fore, as a circle described about the point in which the star 
would be seen from the sun. Moreover, the star's apparent 
place, and the earth's real place in their respective orbits would 
be diametrically opposite. 

At a point between the plane of the ecliptic and its pole, 
the parallactic orbit would be an ellipse, the ratio of whose 
axes would depend on the latitude of the star. 

390. Discovery of annual parallax. — It is justly reckoned 
among the greatest achievements in practical astronomy, that 
the annual parallax has, in a few cases, not only been clearly 
detected as existing, but has been satisfactorily measured y 
though it is never so great as 1". 

The parallax of a Centauri is §".§1 ; that of 61 Cygni, Q".o5; 
of a Lyras, 0".26 ; of Sirius, V .23. A few others have been 



DISTANCES OF STARS. 199 

obtained, which are still smaller, and therefore less relia 
ble. 

The parallax of a star is most satisfactorily determined, wher 
it is in the same telescopic field with other stars. For then 
the distances between the stars may be measured with greal 
precision by a micrometer, and all errors arising from aberra- 
tion, refraction, and instrumental disturbance are wholly 
avoided, because all the stars in the same field are affected 
alike by these causes of displacement. Parallax is the only cir- 
cumstance which can produce an annual change in their rela- 
tive positions. The star 61 Cygni is, in this respect, very 
favorably situated, and its parallax is thought to be quite 
accurately determined. 

391. Distances of those stars whose parallax is known. — 
If a triangle is formed by the lines joining the sun, earth, and 
star, and the angle at the sun be a right angle, we have the 
proportion 

Sin an. par. : rad : : 92,381,000 miles : dist. of the star. 
This gives the distance of a Centauri, the nearest star, 
21,000,000,000,000 miles, nearly. Light, moving at the rate 
of 185,000 miles per second, would require about 3.6 years to 
come from that star to us ; 9.3 years from 61 Cygni ; 12.6 
years from a Lyras ; and 14.2 years from Sirius. And if we 
reckon the parallax of the pole-star at 0".07, as it has been com- 
puted to be, it requires 47 years for its light to reach us. 

In order to compare these amazing distances with the dimen- 
sions of the solar system, we may use with advantage the dia- 
gram described in the note, Art. 263. The distance from the 
sun to Neptune being represented by 30 feet, the distance of 
the nearest star, «. Centauri, must be represented by 40 miles, 
and that of 61 Cygni by 110 miles, etc. Thus isolated are the 
systems of the universe from each other. 

As to all other stars besides those above named, it is only 
known that they are still more distant. There is no improb- 
ability that, from the remotest telescopic stars yet seen, light 
may occupy thousands of years in coming to us. Therefore, 
we see all the stars as they were years ago , perhaps not aa 
they are now. And if at any time a change has been detected 



200 STARS ARE SUNS. 

in the aspect or place of a star, that change occurred, not when, 
it was seen, but 10, 100, or 1,000 years before, according to its 
distance. 

392. Nature of the fixed stars. — The stars are situated at 
such vast distances from the solar system, that if they merely 
reflected the light of the sun, they would be invisible. In 
order to exhibit such brightness as they do, they must not only 
shed light, but a very intense light of their own. They can. 
not be compared with any one of the bodies in the solar sys- 
tem, except the sun itself. All the fixed stars, therefore, are to 
be considered as suns, and probably the centers of systems re- 
sembling the solar system. It is ascertained, respecting some, 
of those stars whose distance is known, that they shed more 
light than the sun. For example, a Centauri has been found 
to shed near four times as much light as the sun. For the 
light of the sun at the earth is about 500,000 times as great as 
the light of the full moon. And the light of the full moon was 
found by Sir John Herschel's observations to equal 27,000 
times that of a, Centauri. Therefore, the light of the sun at 
the earth is (500,000 x 27,000) 13,500,000,000 times that of 
a Centauri at the earth. But that star is 230,000 times as far 
off as the sun. And since the quantity of light received from 
a luminous body varies inversely as the square of the distance, 
if a Centauri were brought as near to us as the sun, its light 
would be 52,900,000,000 (= 230,000) 2 times as great as it is at 
present, or nearly four times as great as the light of the sun. 

In a similar manner, Sirius, the brightest, but not the, 
nearest fixed star, is found to shed 100 times as much light as 
the sun. 

On the other hand, if the sun were removed from us to the 
nearest fixed star, its apparent diameter would be only T ^ ", and, 
therefore, would be a star having no sensible magnitude, and 
having only J of the brightness of Sirius. Appendix J. 

393. Proper motion of the stars. — There is increasing evi- 
dence that there is among the stars a parallactic motion of a. 
higher order than the annual parallax already noticed. The 
entire solar system appears to be moving toward a certain. 



DOUBLE STARS. 201 

point in the constellation Hercules, whose right ascension is 
260°, and its declination 35° north. This motion of the system 
is inferred from what is termed the proper motion of the stars. 
Since the time of Hipparchns (130 B. C), Sirius, Arcturus, 
and Aldebaran have changed their position southward more 
than half a degree. The star 61 Cygni moves 5" each year, 
\i Cassiopeiae 4", and s Indi 8" ; and a large number of other 
stars have a small progressive motion. The general effect of a 
motion of our own system would be to cause a minute ap- 
parent separation of the stars in the region toward which we 
are moving, and a crowding together of the stars in the region 
from which we move. From a comparison of the proper mo- 
tions of several hundreds of stars, a motion of the solar system 
in the direction named above has been deduced. And the rate 
of that motion has been estimated to be about 154,000,000 
miles per year, which is only one-fourth the earth's velocity in 
its orbit. 

If the motion is really perceptible, it is probable that a 
change of direction will, after a few centuries, manifest itself, 
from which something may be inferred as to the position and 
magnitude of the orbit which the sun describes. 

Some of the stars have a proper motion, which can not be ex- 
plained by the supposed motion of the solar system. In those 
cases, it must be concluded that they are themselves describing 
vast system-orbits about some distant center. Appendix K. 

394. Double stars. — It is discovered in a great number of 
instances that a fixed star, when examined by the telescope, 
really consists of two stars, very close to each other. If the 
distance between them does not exceed 32", such stars are 
called double stars. Their distance apart is often less than 1", 
and some are so close, that the highest power of the telescope 
and the most acute vision are requisite to separate them. 
Hence, certain double stars are habitually used as tests of the 
excellence of an instrument. 

When Sir William Herschel first began his observations on 
this class of objects, in 1780, he knew of only four ; but he ex- 
tended the list to 500 himself, and the number now known ex- 
ceeds 6.000. 



202 DOUBLE STAKS. 

395. Relative intensity and color. — In comparatively fe^ 
instances are the two stars equally bright. They sometimes 
differ so little as to fall within the limits of the same magni- 
tude; but geuerally they are of different magnitudes. Thus, 
the component stars of y Leonis are of the 2d and 4th magni- 
tudes ; of 7] Lyrse, 4th and 8th ; and of the pole-star, 2d and 
9th. Figures 1, 2, 3, and 4, in PL III., present the telescopic 
appearance of the double stars there named. In 4, they are so 
close as to appear like a single star, of tapering form. 

A fact of great interest, in relation to double stars is, that 
they often differ in color. Sometimes these colors are com/pie- 
mentary / that is, they are such as would compose white light, 
if mingled together. In such cases, if the stars differ much in 
magnitude, the appearance of color in the fainter star may be 
only an illusion. But this can not be true when the colors are 
not complementary. The components of y Andromedee are 
orange and green ; of % Bootis, white and violet ; of a Herculis, 
yellow and blue ; and of (3 Scorpionis, white and blue. 

Single stars are frequently of a deep red color ; but a decided 
case of green or blue is never met with, except in a component 
of a double star. 

396. Two ways in which stars might appear double. — The 
two stars which compose a double star may be supposed either 
to be really near each other, or only to appear near together, 
because they fall almost into the same line of vision, while one 
is actually at an immense distance beyond the other. In the 
latter case, the stars are said to be optically double. When Sir 
William Herschel commenced examining double stars, he very 
naturally supposed that, in the very few cases known, one star 
happened thus to be nearly in the same visual line with the 
other; and he began the work of observing them, with the ex- 
pectation of detecting annual parallax in objects so favorably 
situated. For, if the nearer star is perceptibly affected by par- 
allax, it would exhibit an annual motion relatively to the more 
distant star, in a manner not to be mistaken. 

397. Binary stars. — It soon became evident, however, that 
double stars are too numerous to allow the supposition that 



ORBITS OF BINARY STARS. 203 

their apparent proximity is only casual. It was calculated 
that the chance, that of all the stars visible to the naked eye, 
two would accidentally appear within 4" of each other, was 
only 1 in 9,000 ; whereas one hundred such cases were already 
known. 

But another most interesting discovery was presently made ; 
namely, that some of the double stars exhibit motions which 
indicate a revolution of one around the other — or, rather, of the 
two around a common center, and in periods of various lengths, 
having no connection whatever with the earth's annual motion. 
Such motion can not be parallactic ; it must be real ; and such 
stars are not optically, but physically double. They are called 
binary stars, and are to be regarded as the centers of double 
stellar systems. 

398. Gravitation outside of the solar system. — The binary 
stars afford evidence that the same law of attraction which pre- 
vails within the boundary of the solar system prevails also at 
immeasurable distances beyond it. In the case of every binary 
star which has yet completed the whole, or any considerable 
part of its revolution, since its discovery, it is found that the 
path of one component star is an ellipse, while the other occu- 
pies one of the foci within it. Hence, the law of attraction is, 
gravity varies inversely as the square of the distance, just as 
within, the solar system. Though the relative motion may be 
represented by considering either star as occupying the focus, 
and the other star as revolving about it, yet the true focus is 
the center of gravity between them, while each describes its 
orbit about that center. 

399. The real and the apparent orbit. — It is not to be as 
sumecl that the plane of a stellar orbit is perpendicular to oui 
line of vision. But if it is oblique, although it is always pro- 
jected on the sky as an ellipse, yet the apparent eccentricity 
may differ in any degree from the real eccentricity, and the 
central star will probably appear out of the focus of the ap- 
parent orbit. The true orbit, however, can be readily deduced 
from the apparent one, by means of the position of the central 
star. If the plane of revolution of a binary star were coinoi- 



204 



PERIODS OF BINARY STARS. 



Fig. 88. 



dent with oar line of vision, one star would appear to oscillate 
in a straight line across the other. 

The ellipse, BCD (Fig. 88), represents the apparent orbit of 
£ Ursse Hajoris, the central star 
being at A. The real orbit, of 
which A is the focus, is BDF. 

The apparent orbit of a Centauri 
is still more eccentric (Fig. 89), 
compared with the real one, be- 
cause more oblique to the Hue of 
vision. It has not yet described '/ 
quite half its orbit, since it began 
to be observed. 

At the bottom of PI. III. are 
shown the relative positions and 
distances of y Yirginis from 1837 
to 1860, and the form of the apparent orbit. The real orbit 
is even more eccentric, the major axis being somewhat fore 
shortened by obliquity. 

Fig. 89. 





400. Periods of Unary stars. — The shortest period knows 
is that of £ Herculis, about 31 years. The period of n Coronse 
is 43 years ; that of I Ursas Majoris (Fig. 88) is 58 years. 
These, and a few others of short period, have completed their 



PERIODS OF BINARY STARS. 205 

revolutions once or twice since they were discovered. The 
orbits of such are quite accurately determined. One revo- 
lution of a Centauri (Fig. 89) has not yet been made since its 
discovery ; its period is calculated to be 77 years. A large 
number of binary stars, whose periods are computed to be some 
hundreds or thousands of years, have been observed as yet only 
through a short arc ; hence their periodic times, and the forms 
of their orbits, are quite uncertain. 

401. Dimensions of stellar orbits. — There are two binary 
stars whose parallax has been so satisfactorily measured, that 
their distances from us may be considered as well known • 
these are a Centauri and 61 Cygni. Hence, by the angular' 
length of the semi-major axes of their orbits, we may find the 
mean radius vector of each. The major axis of the orbit or 
a Centauri is about 30 ", and its distance from the earth is 
21,000,000,000,000 miles. 

.-. rad : sin 15" : : 21,000,000,000,000 : 1,464,000,000 miles ; 
which is equal to about 16 times the earth's distance from the 
sun. The distance between the components of 61 Cygni is 
about 4,012,000,000 miles. 

402. Masses of the binary stars. — For those binary stars 

whose periods and distances apart are known, the mass of the 

J33 
system can be computed. For M oo — ; hence, for a Centauri 

(the earth's distance from the sun, and its period being called 1), 

16 3 
M = ■— - = 0.69. That is, the mass of the two components of 

a Centauri is about 0.7 of the mass of the sun and earth. So, 
for 61 Cygni, whose period is computed to be 540 years, and 
the distance of the two components 44 times the radius of the 
earth's orbit, the mass of the double star is 0.3 of the mass of 
the sun and earth. 

403. Triple and quadruple stars, — There are a few in- 
stances of three or four stars, which are known to be physically 
connected, and to constitute a system. Figs. 5, 6, PL III., 
present the appearance of 11 Monocerotis and C Cancri. In 



206 PEKIODIC STARS. 

the latter, the two close components revolve in 59 years, and 
the distant one more slowly. The faint star e Lyrae is quadru- 
ple, consisting of two very close double stars. They give evi- 
dence of belonging to one system, but their revolutions are ex- 
ceedingly slow. 

404. Periodic and temporary stars. — There are among the 
fixed stars several instances in which there appear to be revolu- 
tions of another sort, the nature of which is not understood. 
Stars which exhibit these changes are called periodic stars. A 
remarkable example occurs in the star o Ceti. It passes 
through its changes of brightness in about 11 months. When 
brightest, it is of the 2d magnitude, and remains so for twc 
weeks. It then diminishes during 3 months to the 10th mag- 
nitude, remains thus 5 months, and increases again during 3 
months to its maximum of brightness. 

Algol (/3 Fersei) has a very short period, occupying only 2d. 
20h. 48m. Its changes succeed each other with great regular 
ity, thus : 

During 2d. 14h. Om. it remains of the 2d magnitude. 
" Od. 3h. 24m. diminishes from 2d to 4th. 
" Od. 3h. 24m. increases from 4th to 2d. 



2d. 20h. 48m. whole period. 

Some of this class of stars have periods of only a few days, 
while in others the changes go on very slowly, and appear to 
require several years. The periods of some are quite uniform, 
and of others irregular. As accurate observations are mul- 
tiplied, the number of known periodic stars is constantly in- 
creasing. 

To this class probably belong those stars which are called 
temporary stars. That of 1 572 is celebrated. It appeared so 
suddenly, and of such brilliancy, as to attract the attention of 
common people, and rapidly increased, till in a few weeks it 
surpassed Jupiter in brightness. It then faded slowly, and 
after about 1^ years entirely disappeared. Several other cases 
less marked than this are on record. And the earlier cata- 
logues contain numerous stars which are not to be found at the 
present day. Undoubtedly some of these records are mistakes, 



NEBULA. 207 

£11 two or three instances, it is known that the bodies were 
planets, not fixed stars. But in the course of coming centuries, 
some of the temporary stars may again become visible, and 
thenceforward be recognized as periodic stars. 

405. Cause of periodicity. — The conclusion can not be 
avoided, that the variable magnitudes of stars, at least when 
they recur regularly, are the result of some sort of revolution 
More than this is mere conjecture. In some cases, the star 
may be partially dark on one side, and produce the changes 
by rotation on its axis. In others, there may be opaque bodies, 
either single or existing in groups or zones, revolving about 
the central star. 

Newton suggested that the sudden appearance of a tem- 
porary star might be the result of a comet falling upon the 
central body, which was before invisible, and causing confla- 
gration. 

406. Clusters of stars. — The fixed stars are frequently 
grouped together in clusters, such as the Pleiades, in Taurus ; 
Presepe, in Cancer ; and Coma Berenices. If a telescope of 
low power is used, the number of stars appears greatly in- 
creased. Figure 1 in PL IV. gives a telescopic view of the 
Pleiades. 

There are others which to the naked eye appear nebulous, 
but by the use of the telescope are plainly seen to be clusters ; 
and in some of them the stars are so numerous as not to be 
easily counted. The clusters in Perseus and Hercules are fine 
examples. For the latter, see PI. IY., Fig. 3 ; a is its appear- 
ance with a low power ; b is the central part of it with a high 
power. 

407. Nebula. — These are faint patches of light, having gen 
erally an ill-defined edge, and in ordinary telescopes presenting 
the same nebulous aspect which the closer clusters do to the 
naked eye. As the powers of the telescope are increased, 
many nebulae are resolved into clusters of stars, while many 
others retain their nebulous appearance under every power yet 
employed. The number of nebulae now known exceeds 5,400, 



208 FORMS OF NEBULA. 

Their forms are exceedingly various ; and in some cases they 
seem in this respect to be greatly changed as the telescope is 
improved in its magnifying and defining powers. 

Since every advance which is made in the Construction oi 
instruments resolves some nebulas into clusters of stars, many 
astronomers have been led to suppose that all nebulas are clus- 
ters, only too remote to be resolved by means hitherto em- 
ployed. Some facts, however, connected with this class oi 
bodies seem to indicate that there are, in some regions of space, 
immense tracts occupied with nebulous matter not yet formed 
into stars. 

408. Varieties of for in among nebula. — 

1 . Globular. A large number, especially of the smaller neb- 
ulas, present a circular outline, and grow brighter gradually 
from the circumference toward the center, thus suggesting the 
idea of a spherical form. The nebulous stars, so called, differ 
from them in that the nebulosity continues nearly uniform up 
to a central star. The planetary nebulae have a well-defined 
edge, and no bright center, and therefore bear some resem- 
blance to a planet. 

2. Elliptical. Several nebulas present the appearance of an 
oblate spheroid seen edgewise. The most remarkable example 
is the great nebula of Andromeda. Its length is 1^°, and it is 
easily seen by the naked eye (PL IV., Fig. 2). The dumb- 
bell nebula, between Cygnus and Aquila, appears in the best 
telescopes to have an elliptical shape. The brightest part of 
it has a form slightly resembling a dumb-bell, or an hour-glass. 
(PI. II., Fig. 4). 

3. Spiral. This description of nebulas is becoming rather nu- 
merous since the latest improvements in telescopes. Some 
nebulas of very irregular shape, as formerly described, exhibit, 
in the best instruments of this day, delicate appendages having 
a spiral arrangement. The whirlpool nebula, near the tail of 
Ursa Major, is the most remarkable instance of this form (PL 
IV., Fig. 5). The crab nebula, in Taurus, may yet be found to 
Delong to this class (PL IT., Fig. 4). 

4. Annular. A few nebulas have an outline nearly circular 
or elliptical ; but appear more luminous on the edges than in 



MAGNITUDES OF NEBULAE. 209 

the central part. Such are called annular nebulae. The ap- 
pearance is that of a hollow sphere or spheroid ; in which case 
we look through the greatest depth near the edges. An inte- 
resting example is situated in Lyra, midway between (3 and y 
(PL IL, Fig. 3). 

5. Irregular. Besides the foregoing forms, which are all in- 
dicative of a central force, and of revolution, there are various 
shapes of great irregularity. None is so celebrated as the great 
nebula of Orion, which has been a subject of observation 
and record for more than two centuries. It becomes more ex- 
tended and more complex with every new improvement in tel- 
escopes. 

409. Magnitude of clusters and nebulce. — Every cluster of 
stars, whether a complex system of suns or not, must occupy 
an immense space. They are at least as far distant as the 
nearest star, and how much further we can not know, and yet 
they fill a sensible angle, and some of them a large one. It is 
easy, therefore, to assign the lowest limit for their dimensions. 
The length of the nebula in Andromeda is 1|°. Supposing it 
as near as a Centauri, its absolute length must be 6,000 times 
the distance from the earth to the sun. And if it be many 
times further from us than the nearest star, which is far more 
probable, then its dimensions must be just so many times 
greater. 

410. Changes in the nebulce. — In repeated instances it has 
been thought that the forms of certain nebulas had essentially 
altered since their discovery. But this is not certain ; for it is 
found that the same nebula assumes a new aspect as the tele- 
scope is improved, because some of the more delicate features, 
which were not before noticed, are brought to view. It may 
be, therefore, that all apparent changes of form hitherto noticed 
are to be explained in this way. 

But there are a few faint nebulas, which are known to have 
grown more dim within a short time ; for they can not now be 
seen by the same instruments which only a few years ago 
brought them distinctly into view. In one or two in- 
stances, a nebula has entirely ceased to be visible. Such 

14 



210 THE GALAXY. 

bodies may, perhaps, have regular changes, like the periodic 
stars. Appendix L. 

41 1. The galaxy. — This is a belt or zone, of nebulous ap- 
pearance, which encircles the heavens, nearly coincident with 
a great circle, and cuts the plane of the equator at an angle of 
63°. It is usually called the milky-way. 2s~ear the constella- 
tion Cygnus, it divides into two parts, which continue separate 
nearly a semicircle (150°), and then reunite. Its edges are 
generally ill-defined, and also quite crooked and irregular T 
having many projections and indentations. 

The telescope shows that the whiteness of the galaxy is due 
to unnumbered stars, too faint to be seen individually. Their 
distribution is quite unequal ; the stars, in some parts, being 
crowded very closely together, while here and there spaces oc- 
cur which contain but few. These inequalities are most marked 
in the southern hemisphere. A small portion of the southern 
galaxy is shown in PI. III. In the most luminous parts, Sir 
William Herschel estimated that, within an area less than z ±q 
part of the hemisphere, there passed the field of his telescope 
50,000 stars, large enough to be distinctly seen. The whole 
number of stars in the milky-way is to be reckoned by millions. 

It appears, therefore, that by far the largest part of the stars 
which are within the reach of our vision lie in a thin stratum 
or ring, in the plane of which the sun is situated. As we our- 
selves, being near the sun, are in this plane, we see the stars 
mostly crowded into the zone or belt which is called the 
galaxy, while over the other parts of the sky they are more 
sparsely distributed. 

412. The nebular hypothesis — What it proposes. — The hy- 
pothesis which is known by the name of the nebular hypothesis 
proposes to explain in what manner the bodies composing the 
solar system may have arrived at their present state, as to mo- 
tion, condition, and mutual relations, through the operation oi 
known laws, which the Creator has employed during the 
almost countless ages since the material was at first formed. 

413. Argument from analogy. — The organized bodies on 



THE NEBULAE HYPOTHESIS. 211 

the earth, whether animal or vegetable, are not created in their 
mature and perfect state, performing at once all the functions 
for which they were designed; but they grow to this condition 
by a series of changes, which extend generally through a num- 
ber of years. 

So the soils of the earth were not first formed in their present 
condition, fitted to sustain the vegetation which clothes them ; 
but are the result of slow disintegration of the rocky mountain 
tops, through the action of water and changes of temperature. 

It is more in accordance with the Creator's plan of operation, 
so far as we can discover it, that the sun, planets, and satellites 
should have been brought into their present condition through 
a long-continued course of change, than that they should have 
been created and set in motion as we now see them. 

414. Facts in the solar system which form the oasis of the 
hypothesis. — 

1. The sun, the planets, and the satellites, so far as they are 
known to rotate at all on their axes, rotate nearly in the same 
direction, from west to east. And the revolutions of all planets 
about the sun, and of all satellites about their primaries, with 
but few and trifling exceptions, are in the same general direc- 
tion, from west to east. 

2. The sun, which contains nearly the whole material of the 
system, is a sphere in a condition of intense heat. The interior 
of the earth is in a red-hot melted state, as is proved by the 
volcanoes on its surface. The moon is covered with volcanic 
craters, which show that it is, or has been, in the same condi- 
tion, internally, as the earth now is. 

415. The nebular hypothesis stated. — It assumes that the 
whole space occupied by the solar system, and extending far 
beyond its present limits, was filled with nebulous matter, in an 
exceedingly rare and intensely heated condition ; and that this 
entire mass was put into a state of rotation in the direction 
which we now call from west to east. 

This assumption being made, the following consequences 
would ensue, during the lapse of immense periods of time, in 
accordance with the we 1-known laws of the material creation. 



212 THE NEBULAE HYPOTHESIS. 

By gravity and the centrifugal force, the vast nebula takes a 
spheroidal shape. 

Heat is radiated from its exterior into the boundless space 
around it ; and by this loss, the nebula contracts in diameter. 
But as it contracts, the given velocity of rotation at the surface 
causes a quicker rate of revolution, until, at length, the cen- 
trifugal force of the equatorial part equals the attraction toward 
the center of the entire mass. As soon as these two forces are 
equal, the equatorial part rotates independently of the interior, 
while the latter contracts still further, and leaves the superfi- 
cial part revolving as a nebulous ring. 

After the central portion has left the ring, it goes on con- 
tracting as before, till it leaves a second ring. Thus, an indefi- 
nite number of concentric nebulous rings may be left, each 
revolving from west to east, and at a swifter rate according as 
it is nearer the center. The central mass, which thus succes- 
sively deposits its rings, is the sun of the system. 

416. "While the material composing each ring goes on cool 
ing and contracting, unless the quantity is exactly equal on 
every side, which is improbable, the whole of it, at length, is 
drawn toward the heaviest side, until it is gathered into a 
spheroid, revolving once on its own axis, while it revolves once 
around the central mass. These spheroids ire the planets, re- 
volving around the sun. 

But as the planetary spheroid continues to contract by cool- 
ing, its rate of rotation is quickened, untr it leaves its equa- 
torial part revolving in a ring about it, in the same manner as 
the central nebula has done ; and this it may do in repeated 
nstances. 

These subordinate rings are likely also to collect into so 
many spheroids, revolving about the larger ones, and on their 
own axes. These are satellites. In case the parts of a ring are 
very exactly balanced, they may preserve their condition of a 
ring, instead of gathering into a satellite. Ax. example is seen 
in the ring of Saturn. 

It is conceivable that a multitude of &u*all rings, instead ol 
one large one, may be detached from the central mass when 
the separation occurs. This seems to have been the c^se in 



THE NEBULAR HYPOTHESIS. 213 

the formation of those rings from which the planetoids were 
formed. 

417. After the planets and satellites have cooled sufficiently, 
they become non-luminous bodies, and are gradually changed 
from nebulous into a liquid or solid condition. And, in a 
given case, the exterior may be solid, while the interior re- 
mains in a liquid and highly heated condition. This is the 
present state of the earth, and the present or recent condition 
of the moon. 

That the planes of motion throughout the system are not co- 
incident, is to be ascribed to disturbing influences which the 
several bodies have been exerting on each other during the 
vast periods of time that have elapsed since they were detached 
from the solar mass. 

418. Application to other systems. — Every fixed star which 
is single may be the condensed nucleus resulting from an op- 
oration similar to that which has been described; and the 
double and triple stars may be considered as cases in which 
either the nebula became divided into two or three parts, be- 
fore the contraction had proceeded far, or else the nebulous 
mass, being very oblate, a large part of it was detached at 
once, and collected into a body, nearly equal to the central 
part. 

The nebulae of regular form, not capable of being resolved 
into separate stars, may still be in the condition of the solar 
system before its rings began to be separated from the original 
bodv. 



APPENDIX 



A.— Art. 107. 

The exact period of the sun's rotation is not easily deter- 
mined, because of the independent motions of the spots them- 
selves. That they do have such motions is apparent from the 
fact that they differ from each other somewhat in their east- 
ward velocity, and also that some of them move a little north- 
ward or southward wmile crossing the disk. Among the various 
results obtained by different observers, the lowest is about 
25 days, and the highest about 25 days and 12 hours. 



B.— Art. 111. 

The perspective effect described in Art. Ill, may perhaps 
be better understood by the aid of Fig. I., which represents a 




section of the sun through a spot. Let ah be the breadth of 
the opening in the outer stratum, cd that of the narrower one 
in the inner stratum. When this spot is seen near the middle 



APPENDIX. 



215 



of the disk, we look into it almost at right angles to the sur- 
face, along the lines marked p, p, and can see ef of the denser 
part of the sun (which is the macula), and also some of the 
inner stratum on all sides of cd / and this is the umbra. But 
when the spot is very near the edge, we look along the lines 
rr through ab, and can see only that part of the inner stratum 
which is beyond cd; in other words, only that part of the um- 
bra is seen which lies nearest to s, the edge of the disk. 

C.— Art. 112. 

The bright points and streaks which are generally visible 
over most of the sun's disk, giving it a mottled appearance, 
are called f amice (little torches), and the dark specks among 
them are often called pores. The faculse are described by some 
observers, as having the appearance of willow leaves crossing 
each other in all directions, and by others, as resembling rice 

Kg. II 













grains, or bits of straw. They are most conspicuous at the 
edges of spots, and at places where spots are forming or closing 
up. Irregular bands of faculse are frequently seen projecting 
themselves with great velocity over the area occupied by a 
spot, and even forming bridges entirely across it. Fig. II. 
imperfectly represents these appearances. 



D.— Art. 115, and 228. 

The combination of the spectroscope with the telescope has 
enabled astronomers to gain considerable additional knowledge 



216 APPENDIX. 

respecting the nature and condition of the sun's exterior. See 
Nat. Phil., Art. 398-400. 

The dark lines of the solar spectrum show that the photo- 
sphere consists of the following substances in the gaseous 
state: sodium, calcium, magnesium, chromium, iron, copper, 
zinc, barium, nickel, hydrogen, etc. The intense light of the 
liquid parts below, shining through these gases, causes their 
spectrum lines, which would otherwise be bright colored lines,, 
to become dark ones. 

The same instrument has more recently proved the existence 
of a less luminous envelope outside of the photosphere. The 
appearance of irregular projecting masses of faint red light 
from behind the moon during a total solar eclipse, had pre- 
viously led to the suspicion of such an atmosphere. See 
Art. 228, 3. By the use of modern instruments, not only can 
these protuberances, or prominences of reddish light be viewed 
at any time, but also the envelope itself can be traced entirely 
around the disk of the photosphere. This outer covering is. 
called the chromosphere, because its spectrum exhibits colored 
instead of dark lines. This covering of red-hot gas consists 
largely of hydrogen, having an average depth of several hun- 
dred miles ; but, being generally in a state of extreme commo- 
tion, its more elevated parts are from 50,000 to 100,000 miles, 
high. The parts thus thrown upward by the terrific forces in 
operation there, are sometimes completely detached from the 
rest. The prominences of the chromosphere often resemble 
mountains, trees, flames, or clouds ; but more frequently they 
assume fantastic forms wholly indescribable. These forms 
change very rapidly, indicating a motion of several thousands 
of miles in a single hour. In some cases there is evidence of 
rotary motion parallel to the surface of the sun — that is, there 
are vast whirlwinds of fire. In others, jets of red-hot hydro- 
gen are spouted upward to the height of 50 or 60,000 miles. 
Fig. III. will convey some idea of the variety and singularity 
of the forms of the prominences of the chromosphere. The 
photosphere, or bright surface of the sun, is represented in each 
part of the figure by the curve ah. 

The corona, which is white, and surrounds the chromosphere, 
extends considerably beyond its highest prominences. It is 



APPENDIX. 



217 



distinctly seen on- 
ly during the to- 
tality of a solar 
eclipse, and its 
boundary is not at 
a uniform height 
on all sides, but 
varies irregularly 
from 100,000 to 
200,000 miles in 
height from the 
photosphere. 

A still fainter 
white light is seen 
during the time of 
a total solar eclipse, 
extending outward 
beyond the coro 
na ; this is called 
the halo. It was for 
a time suspected 
to be an effect pro- 
duced by our own 
atmosphere. But 
there is increasing 
evidence furnished 
by recent eclipses, 
that it truly sur- 
rounds the sun. 
Its extent is very 
unequal on differ- 
ent sides, having 
in some places 
deep gaps reach- 
ing down to the co- 
rona, and in other 
parts extending 
upward to nearly 
twice the diameter 
of the sun. 



Fig. III. 






218 APPENDIX. 

E.— Art. 116. 

There is an interesting connection between the periodicity 
of the solar spots, and that of magnetic disturbances on the 
earth. By a careful comparison of these phenomena, as ob- 
served and recorded through a period of nearly a hundred 
years, it is found that with the periodic increase and decrease 
of the amount of spot-surface on the sun's disk, there is a cor- 
responding increase and decrease of terrestrial magnetic storms, 
indicated by the agitations of the needle and the occurrence 
of the aurora borealis. In each case, the maximum occurs at 
about the same time once in ten or eleven years, and the mini- 
mum at nearly corresponding times between. In general, 
there are frequent auroras and frequent disturbances of the 
needle in those years in which the sun exhibits the greatest 
spot-area ; and when the solar spots are few and small, auroras 
are infrequent, and the needle is but little disturbed. "What- 
ever may be the cause of spots on the sun, it is in the highest 
degree probable that the same cause produces these alternations 
in terrestrial magnetism. 



F._ Art. 256. 

There appears to be increasing evidence that there is at 
least one planet revolving within the orbit of Mercury. A 
round, dark spot has been repeatedly seen crossing the sun's 
disk ; and a comparison of the dates of such observations leads 
to the belief that an inferior planet exists, w T hose periodic time 
is about 39 days. The name Yulcan, has been already given 
to the supposed planet, The existence of such a body, or else 
of a group of smaller bodies, has been for some time suspected, 
because of an unexplained morion of the perihelion of Mer- 
cury's orbit. 

G.— Art. 357. 

It does not necessarily follow from the reasoning in Art. 357, 
that all the light received from a comet is reflected. What is 
there stated would be true if a part is reflected, and another 
part originates in the comet itself. And the spectra of some 



APPENDIX. 219 

comets examined within a few years, furnish quite satisfactory 
evidence, that while they reflect the sun's light, they also 
radiate the light of some incandescent substances, either gas- 
eous, or in the state of a comminuted solid. 

H.— Art. 367. 

Biela's comet, which separated into two parts in 1846 while 
in sight from the earth, and which reappeared as two comets 
in 1852, has since then, it is believed, been partially or wholly 
divided into innumerable cometary fragments. For it has 
failed to appear at the times of its expected return since 1852, 
and in its stead there has been an unusual number of shooting 
stars coming into the earth's atmosphere at times and in direc- 
tions corresponding to such a supposition. The path of the 
earth and that of Biela's comet so nearly intersected each other, 
that if the latter body has suffered the catastrophe supposed, 
it was to be expected that some of its fragments would meet 
the earth, and appear in its atmosphere as shooting stars. 

I.— Art. 380. 

The dissolution of a comet into a group or ring of meteors 
has taken place in other instances besides that mentioned in 
Appendix H. The annual meteoric shower of August 10th, 
comes from a ring which coincides with the orbit of Comet III., 
1862. A small arc of this orbit is represented in Fig. IT., 
intersecting the earth's orbit at A, through which point the 
earth passes on the 10th of August. The planes of the two 
orbits intersect in the line AB, and their inclination, the angle 
ESM, is 64° 3'. The perihelion of the meteoric orbit is P ; 
and PC drawn through the sun S, and produced, is the axis, 
which meets the aphelion at the distance of about 10,000,000,000 
miles from the sun, or more than three times the distance of 
Neptune. The cometary fragments seem to be distributed 
around the whole circuit of the orbit, though unequally ; so 
that the earth, when it passes across their path, always meets 
a few, and sometimes large numbers of them. The small 
comet, III., 1862, was probably the mere remainder of a large 



220 



APPENDIX. 



cornet, which has for ages been scattering its particles along its 
path ; for the August shower has been known for a long time. 



Fig. IV. 




Another example of identity between a comet and a me- 
teoric shower, is that of comet I., 1866, or Tempers comet, 
and the shower of November 13th. The orbits have the same 
elements, and their periodic time is 33 \ years. In this case, 
the fragments of the comet, instead of occupying the whole 
circumference, are gathered into a group, of such length, how- 
ever, that the earth strikes into it on three successive returns 
to the same place in its own orbit. Among the most brilliant 
displays of meteors from this group are those of 1799, 1833, 
and 1867. Fig. V. shows a short arc of this orbit, along which 
the meteors are moving in the direction of the arrows, in a 
group of varying thickness. It intersects the earth's orbit 
at A, the planes of the two being inclined at an angle of 



APPENDIX. 



221 



17° 4:4:', represented by ES1I. The aphelion is about as far 
from the sun as the orbit of Uranus. The meteors in both of 



Mar. V. 




the foregoing orbits have a retrograde motion, as the arrows 
show. Hence, the earth meets them, moving partly in a direc- 
tion opposite to its own motion. 



J.— Art. 392. 

Spectroscopic observations made upon the brighter stars 
reveal the interesting fact, that, like the sun, they have a 
gaseous photosphere containing substances of the same nature 
as some of those existing on the earth. For example, a Taori 
has hydrogen, sodium, magnesium, iron, mercury, and several 
other known elements in its gaseous exterior. Likewise, 
a Orionis, by the dark lines of its spectrum, is proved to have 
a constitution much like that of a Tauri. Hundreds of stars 
of the larger magnitudes have in like manner furnished some 
indications of the elements which compose them. But even 
the brightest stars shed so little light at our immense distance 
from them, that only the most conspicuous lines due to a given 
substance are visible. Yet the very exact coincidence of the 
few lines which can be seen, with those of the corresponding 
terrestrial elements, is considered as conclusive proof that 
these elements enter into the composition of such stars. It is 



222 APPENDIX. 

believed, therefore, that at least the brightest stars have a 
physical constitution similar to that of the sun of our own sys- 
tem. Their light, emanating from the denser central parts, 
passes through a luminous gaseous envelope, and by the dark 
lines thus exhibited, reveals the nature of the envelope. 

There is no reason to suppose that the fainter stars, as a 
class, differ in their constitution from the brighter ones. They 
are too far off, however, to afford us sufficient light for ascer- 
taining their true character by any means which have as yet 
been devised. 



K— Art. 393. 

One of the most remarkable discoveries made by the use of 
the spectroscope is that of the motion of certain stars either 
toward, or from the solar system. A certain wave-length of 
light belongs to each point through the length of the spectrum. 
The waves of the red extremity are longest, and those of the 
violet extremity are shortest ; and there is a regular gradation 
from one to the other. Nat. Phil., Art. 436. Hence, every 
line of the spectrum, since it has a fixed place, indicates pre- 
cisely a certain wave-length corresponding to its location. 
Now, suppose that in the spectrum of a star, some of the 
stronger lines of a substance, — hydrogen, for example, — are dis- 
covered and known by their prominence and general locations, 
to be hydrogen lines ; and suppose again, that when carefully 
examined under a high power, and compared with hydrogen 
artificially heated, that they are slightly displaced toward the 
violet end of the spectrum. This shows that the waves are a 
little shorter than those of hydrogen at rest. Such a displace- 
ment proves, therefore, that either the star is coming toward 
us, or we are approaching it ; and the degree of displacement 
indicates the velocity of approach. A star, on the other hand, 
which shows a displacement of a set of lines from their true 
places toward the red extremity of the spectrum, is thereby 
known to be increasing its distance from the solar system. 
Thus, the star Sirius is discovered to be moving from the sun 
at the rate of nearly 30 miles per second. 

The star may indeed be moving in a direction oblique to the 



APPENDIX. 223 

line joining it and the sun ; but the spectroscopic displacement 
indicates only that component of the motion which is in the 
visual direction. The other component, if it exists, is what 
has been long recognized as the proper motion of the star ; 
that is, its angular change of place, which cannot, however, be 
reckoned as a linear quantity, till the distance of the star from 
the solar system is ascertained. 

L.— Art. 410. 

So long as successive improvements in telescopes led occa- 
sionally to the resolving of a nebula into a cluster of stars, it 
remained uncertain whether all nebulae consist of separate 
stars or not, until a new mode of investigation was discovered. 
Notwithstanding the great difficulties in the way of examining 
such faint objects with the spectroscope, the general question 
seems to be satisfactorily answered. All nebulse, which have 
been hitherto resolved, exhibit a spectrum, apparently con- 
tinuous, though dark lines too delicate to be discerned may 
exist. The bodies composing such nebula?, therefore, consist 
of solid or liquid matter, which may or may not be surrounded 
by a gaseous envelope. Also, several of the nebulas not yet 
resolved, show the same kind of spectrum ; which indicates 
that these, too, are solid or liquid, but so remote as not to yield 
to the power of any telescope yet applied to them. 

On the other hand, the larger part of irresolvable nebulas, 
bright enough to be examined, form a spectrum which consists 
only of one, two, or three bright lines ; and these generally 
coincide with those of some known gas. Thus, the ring 
nebula of Lyra gives a spectrum of one line, and that the 
brightest nitrogen line ; and the great nebula of Orion, a spec- 
trum of three lines, — one nitrogen, one hydrogen, and the third 
unknown. 

M.— Aet. 264. 

To identify any Planet, — When the observer has not the use 
of an Ephemeris he can find the approximate, place of any of the 
primary planets by the following process : 



APPENDIX. 



224 



Discarding the elliptical orbit, assume that the planet moves 
uniformly in a circular orbit. Multiply the mean daily motion 
both of the planet and the earth, as given in column VI, Table 
II, page 228, by the number of days that have elapsed since 
the beginning of the century, being careful to include the leap- 
days ; to the product add the corresponding numbers in column 
VII. Divide by 360, so as to reject all the completed revolu- 
tions, and the remainders will be the mean heliocentric longi- 
tudes of the planet and of the earth. 

In Fig. VI, let S be the sun ; E, the earth ; V, the vernal 
equinox as seen from the earth ; V, the same, as seen from the 
sun ; and P the planet, the ecliptic being in the plane of the 
diagram. V'SP will be the heliocen- 
tric longitude of the planet ; V'SE the 
heliocentric longitude of the earth ; 
PSE the difference between them, 
and VEP the geocentric longitude of 
the planet. 

In the triangle SEP, knowing SP 
and SE, either in astronomical units 
or in miles, and the angle ESP, we 
can compute the angle SEP by plane 
trigonometry. As SV and EV are 
parallel, SEV is the supplement of 
V'SE. By subtracting SEV from 
SEP, we have VEP, the required East 
geocentric longitude of the planet. 

If the diagram be held in the plane of the ecliptic, and the 
line EV pointed toward the vernal equinox, EP will point 
nearly in the direction of the planet. The inclination and 
eccentricity of the orbits of Mercury and the moon are so great 
that this method cannot be applied satisfactorily to finding 
their places. 




West 



SPECTROSCOPE. 

(Fauth & Co., Manufacturers, Washington, D. C.) 




TABLE L— THE CALENDAR, 



A Tajl* to r "Nd the Day of the Week of ajsty given date, fkom tee Yeab 5000 

TO THE YEAB 2700 OF THE CHBI3TIAN Era. 



Ckmtueibs before Christ. 










Centuries after Christ 


4SO0 
4100 
3400 
2700 
2000 
1300 
600 


4700 
4000 
3300 
2600 
1900 
1200 
500 


4600 
3900 
3200 
2500 
1800 
1100 
400 


4500 
3800 
3100 
2400 
1700 
1000 
300 


4400 
3700 
3000 
2300 
1600 
900 
200 


5000 
4300 
3600 
2900 
2200 
1500 
800 
100 


4900 
4200 
8500 
2S00 
2100 
1400 
700 



New j 
Style 1 

Old J 

Style ] 

I 


1700 
2100 

0- 

700- 

1400- 

2100- 

C 

B 




1800 
2200 

200- 
900« 
1600' 
2300- 


300- 
1000- 
1700- 
2400* 

F 


1500 
1900 
2300 

400* 
1100* 
1800« 
2500* 

~G~ 


1600. 
2000' 
2400* 

500» 

1200« 
1900* 
2600* 


600« 
1300« 
2000* 
2700' 

~b" 

A 

g" 

F 

d" 

C 
B 


100« 

800* 
1500- 
2200* 


C 

D 
F 
G 
A 
B 
D 
E 
F 
G 
B 
C 


D 


E 


F 
G 
B 


G 


A 


B 

c 



•1 


28- 


56- 
•57 


84- 


D 


E 


A 

G 
F 
E 
C 


E 


F 


A 


B 


•29 


•85 
86 


C 


D 


E 
D 
C 
A 


F 


G 
A 
B 


A 


C 


D 

E 


E 


2 


30 


58 


A 

G 


B 

A 


C 


E 


B 


C 


D 


F 


3 

4* 


31 


59 


87 


B 


D 


C 
D 


D 

E 


E 


F 


G 


32- 
•33 


60- 
•61 


88- 


E 


F 
E 


G 


B 

A 


C 


F 


G 


A 


•5 
6 

7 


•89 


D 


F 

E 


G 


B 
A 


E 
F 

G 


F 


G 


A 
B 


B 


C 


34 


62 


90 


C 


D 


F 


G 


G 


A 


C 


D 


35 


63 


91 


B 


C 


D 
B 


' E 


F 


G 
E 
D 

C 


A 
F 


A 
B 


B 


C 


D 

E 


E 


8- 


36- 


64- 


92- 


G 


A 


C 


D 
C 


A 

C 


C 


D 


F 
A 


•9 


•37 
38 
39 


•65 


•93 


F 

E 


G 


A 


B 

A 


E 
D~ 
C 
A 

G 
F 
E 
C 
B 


D 


E 


F 


G 


10 

11 


66 


94 


F 


G 


B 


D 


E 


F 


G 
A 
B 
D 


A 


B 


67 


95 

96- 


D 


E 
C 


F 
D 


G 


A 


B 

G 


D 


E 

F 


F 


G 


B 


C 


12- 


40- 


68- 


B 


E 


F 

E 
D 


E 


G 


A 

C 


C ! D 


•13 


•41 


•69 


•97 
98 


A 


B 
A 
G 
E 


C 


D 


F 


G 


A 
B 


B 


E 


F 

G 
A 


14 


42 


70 


G 


B 


C 


E 


A 


C 


D 

E 
F 


E 
F 


F 
G 
A 


15 


43 


71 


99 


F 
D 


A 


B 


C 




B 


B 


C 


D 


16- 


44- 


72- 
•73 




F 


G 


A 
G 


C 


D 


E 

G 


G 


B 
D 


•17 
18 
19 


•45 




C 


D 


E 


F 


A 


E 


F 


A 
B 


B 

C 


C 


46 


74 


B 
A 


C 


D 


E 
D 


F 


G 


A 
G 


F 


G 
A 


A 


E 


E 
F 


47 


75 
76- 


B 


C 


E 


F 


G 


B 


C 


D 


20- 


48- 




F 


G 


A 
G 


B 


C 


D 
C 


E 
D 


A 


B 

"d 

E 
F 
G 
B 


C 
E 


D 
F 


E 


F 
A 


G 
B 
C 
D 


•21 


•49 


•77 
78 
79 





E 


F 


A 


B 


C 
D 
E 
F 
A 
B 


G 

A 


22 
23 
24- 


50 
51 
52- 
•53 


D 
C 
A 


E 
D 


F 
E 
C 
B 
A 
G 


G 


A 


B 


G 
B 
G 
F 
E 
I) 


F 
G 


G 
A 


B 


F 


G 


A 
F 
E 
D 
C 


B 


C 


80- 
•81 


B 
A 
G 
F 


D 
C 
B 
A 


E 


A 


B 
D 


C 


D 


E 
G 


•25 
26 


G 


D 


C 


E 


F 


54 


82 


F 


C 
B 


C 


D 


E 


F 


Q 


A 


27 


55 


83 




E 



TABLE I. — THE CALEKDAK. 



227' 




h 

H 
Q 

o 

ffl 

< 


3 J * 1 fc* 


rd 
Eh 


'£ 1 93 


* 1 

GO 


IS 
Eh 


* 


rd 


S 


o3 

GO 


* 1 3 


£ 


rd 

Eh 




03 
GO 


d 

go 


o 



Eh 




E | * 1 1 


o 

3 


a|* 




03 1 S3 
GO 1 GO 


O 



Eh 


£ 


r0 

Eh 


o3 


■ 
GO 


o 


3 

H 


£ 


rd 1 *£ 

Eh 1 Eh 




o 1 o > ! rd 1 'C 

S 1 Eh J !> i Eh J fe 


o3 
GO 


September. 
December. 


CO 


i ill 




io 1 eo 1 t- 1 go 1 os 

« 1 C» 1 « 1 M 1 « 


O 

CO 


i> 


GO 1 OS ! O 1 H 1 OJ 


CO 
CM 


O 


S ! S 


co 1 ^ | io 


CD 


CO 


"tf | iO 


co | ?> 


GO 


OS 








1 


r-l 


cm 


Feb., L.Y. 
August. 


o* 


GO 
O* 


OS 
CM 


O | TM 

CO 1 CO 






o 

O* 


tH I O* 

CM 1 cm 


CO 1 ~* | o 


co 

CM 


CO 
1—1 


tH I IO 

TH 1 tH 


CD 1 J> 1 GO 


OS 


c© 


i> 1 GO 


°I2IS 


CM 

tH 


| H i « | CO 


tH 


m 


1 

►-a 


W 1 © 1 !> 1 00 1 O 
« 1 « 1 « 1 CQ 1 « 


o 

CO 




00 

~~ tH~ 
T-l 


OS 1 O 1 tH I CM 

H 1 « 1 « 1 « 


CO 

cm 


^+1 
cm 


« 1 W 1 ■* 1 o 


CD 


i> 


W 1 «D 1 b» 


GO 


os 


o 

tH 






1 


TH 


CM 


CO 


si 


go 

Oi 


OS 
CM 


O 1 TH 

CO 1 CO 








C3 


CM 1 CO 1 Tt< : IO 

« 1 w 1 « 1 w 


CO 
CM 


CM 


tH 


3 1 S 1 £ 1 s 


OS 


o 

CM 


£- 


00 1 * 1 S 1 s 


CM 


CO 




th | (N | CO 


tH 


iO 


co 


January, L. Y. 
April. 
July. 


0~l tH I 
CO i CO 1 










CO 
05 


^ 1 o 
CM 1 CM 


o 
cm 


CM 


GO 

cm 


o 

CM 


co 


i> 1 CO 

T-t 1 T-t 


OS 
tH 


o 

CM 


cm 


CM 
CM 


OS 


O j H 


CM 


CO 


Tj< 


IO 


05 


CO 


^ 


IO 


CO 


i> 


GO 












1 TH 


February. 

March. 
November. 


35 


CM 


GO 


OS 


o 

CO 


S3 i ! 


OS 


O 
CQ 


CM 


CM j CO 1 ■<* 1 IO 


03 


CO 
tH 


tH 


»o> 


CD 1 I- 1 GO 

H I H I H 


Iffl 


to 


t> 


CO 


OS 1 O 1 T-l 

1 rl 1 H 








tH 


CM | CO | <"* 


January. 
October. 


OS 


o 

CO 


CO 




1 1 


CM 
CM 


CO 
CM 


CM 


cm 


O 1 i> 1 oo 

« 1 « 1 « 


1—1 


co 


*> 

T-l 


GO 


OS 1 © 1 T-l 

H I W 1 « 


GO 


OS 


O 


I—I 


N 1 CO | tJI 


- 


cm 


CO 


-* 


ia | co | t> 



1 B? rf 



C5 

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<1 

CD 
C3 
CM 

m 

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O 
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Ph 

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PQ 

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Eh 

W 
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fe 
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<1 
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<j 

h4 
Ph 

M 
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fa/3 isD 



r3 


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rd 


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£ ^ 



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a s 



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rd' 








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o3 




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rd 
+3 


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c3 


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d 


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228 



ELEMENTS OF THE PLANETS. 

Table II. Elements of the Planets. 



I. 


IT. 


in. 


IV. 


V. 


VI. 


vn. 


vin. 


NAME. 


g 


Relative 


Sidereal 
Revolution 


Synodical 
Revolut'n 


Mean daily 


Heliocen- 
tric Long. 


Heliocen- 
tric Long. 


Mercury. . 


>> 

3 




in days. 


in days. 






Jan.1,1801. 


Jan.1,1885. 


0.387098 


87.969 


115.877 


/ 
4 5 


11 
32.6 


/ 
213 20 


/ 
82 24 


Venus . . . 


9 


0.723332 


224.701 


583.921 


1 36 


7.8 


11 33 


206 54 


Earth . . . 


6 


1.000000 


365.256 




59 


8.3 


100 39 


100 18 


Mars 


3 


1.523691 


686.980 


779.936 


31 


26.7 


64 23 


296 45 


Jupiter. . . 


n 


5.202800 


4332.554 


398.884 


4 


59.3 


112 15 


147 00 


Saturn . . . 


h 


9.538800 


10,759.106 


378.092 


2 


0.6 


135 20 


81 28 


Uranus. . . 


V 


19.183380 


30,686.246 


369.656 




42.4 


177 48 


179 52 


Neptune. . 


5! 


30.054370 


60,228.072 


367.485 




21.7 


227 18 


52 13 


Moon* 


c 


0.000259 


27.321 


29.530 


13 10 


36.6 


118 17 


105 13 





IX. 


X. 


XI. 


XH. 


xni. 


XTV. 


XV. 


XVI. 


xvn. 


a 

02 


Inclinat'n 

of Orbit. 

Jan.1,1801. 


Varia- 
tion in 
100 yrs. 


Eccentri- 
city of 
Orbit, 
Jan.1,1801. 


Variation 
in 100 yrs. 


Long, of 

A. Node 

Jan. 1, 

1801. 

° / 


Motion 
west.in 
100 yrs. 


Long, of 

Perihel. 

Jan. 1, 

1801. 


Motion 
east, in 
100 yrs. 


Rotat'n 

in 
hours. 




/ // 


11 


R.V. 




/ // 


/ 


; // 




3 


7 9 


+ 18 


.205 515 


+ .000 004 


45 57 


13 4 


74 21 


9 43 


24.09? 


$ 


3 23 28 


—5 


.006 811 


-.000 063 


74 51 


32 24 


128 43 


5 12 


23.35? 


8 








.016 792 


-.000 042 






99 31 


19 10 


23.93 


ft 


1 51 6 


-0.2 


.093 307 


+ .000 090 


48 


41 26 


332 23 


26 13 


24.62 


u 


1 18 51 


-23 


.048 162 


+ .000 159 


98 26 


26 19 


11 8 


11 5 


9.92 


h 


2 29 36 


-15 


.056 151 


-.000 312 


111 56 


32 23 


89 9 


32 11 


10.24 


V 


46 28 


+ 3 


.046 611 


-.000 025 


72 59 


59 55 


169 


3 56 


9.50? 


' 1 


1 46 59 




.008 719 




130 6 




46 






•>* 


5 8 40 




.054 908 




13 53 


1934° 


266 10 


4069° 


708.73 


Sun 


















608. 





xvm. 


XIX. 


XX. 


XXI. 


xxn. 


XXIH. 


XXIV. 


XXV. 


.a 
B 


Mean Di- 
ameter in 
miles. 


Mean 

angular 

Diam. 


Relative 
Volume. 


Relative 

Mass. 


RePtive 
D'nsity. 


Rel'tive 
Gr'vity. 


Solar 

Light & 

Heat. 


Velocity 
in Orbit 
in miles 


















per sec. 






11 












2 


2,992 


7 


0.054 


0.065 


1.21 


0.46 


6.67 


29.55 


? 


7,660 


17 


0.880 


0.769 


0.85 


0.82 


1.91 


21.61 


5 


7,918 




1.000 


1.000 


1.00 


1.00 


1.00 


18.38 


$ 


4,211 


9 


0.248 


0.111 


0.73 


0.39 


.43 


14.99 


n 


86,000 


37 


1350. 


311.953 


0.24 


2.64 


.037 


8.06 


h 


70,500 


16 


689. 


93.329 


0.13 


1.18 


.011 


5.95 


w 


31,700 


4 


75. 


14.460 


0.22 


0.90 


.003 


4.20 


f 


34,500 


3 


102. 


16.862 


0.20 


0.89 


.001 


3.36 


€ 


2,161 


1866 


0.020 


0.012 


0.60 


0.16 


1.000 


0.63 


Sun 


860,000 


1924 


1295000. 


326800. 


0.25 


27.71 







Mean geocentric values. 



ELEMENTS OF THE SATELLITES. 



fc^'J 



Table III. Elements of the Satellites. 



THE MOON. 



Mean distance from the earth, (miles) 

Mean sidereal revolution, (days) 

Mean synodical revolution, (days) . . . 

Mean revolution of nodes, (days) 

Mean revolution of apsides (days) . . . 
Mean inclination of orbit to ecliptic. . 

Eccentricity of orbit 

Mean diameter of moon, (miles) 

Diameter, (earth's = 1) 

Surface, (earth's = 1) 

Volume, (earth's = 1) 

Density, (earth's = 1) 

Mass, (earth's = 1) 

Gravity, (earth's = 1) 



238.820 

27.32166 

29.53058 

6793.39108 

3232.57534 

5° 8' 44" 

0.054908 

2161 

0.2730 

■f 3 - or 0.0745 

is or 0.0203 

| or 0.6052 

i-:* or 0.0123 

£ or 0.165 



Satellites 

of 
Jupiter. 



Sidereal 
Revolutions. 



h. m. s. 

18 27 34 

14 36 

42 33 



13 
3 



16 16 31 50 



Distance in 
equatorial ra- 
dii of Planet. 



6.04853 

9.62347 

15.35024 

26.99835 



Distance 

in 

miles. 



260000 

414000 

661000 

1162000 



Diameter 

in 

miles. 



2365 
2123 
3471 
2966 



Satellites 

of 
Saturn. 



22 37 23 

8 53 7 

21 18 26 

17 41 9 

12 25 11 



15 22 41 



21 

79 



25 
7 41 
53 40 



3.3607 

4.3125 

5.3396 

6.8398 

9.5528 

22.1450 

26.7834 

64.3590 



122000 
157000 
194000 
248000 
347000 
804000 
973000 
2338000 



1163 
2908 

1745 



Satellites 

of 
Uranus. 



2 12 29 21 

4 3 28 8 

8 16 56 31 

13 11 7 13 



7.40 
10.31 
16.92 
22.56 



123000 
172000 
282000 
376000 



Satellite of 
Neptune. 



5 21 2 43 



12. 



222000 



230 



MEAN PLACES OF PRINCIPAL STAES. 



Table IV*. Mean Places of Principal Stars ; 1885, Jan. 0. 



No. 



STAR'S NAME. 



Mag. 



9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

21 

25 

2G 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
4(3 
47 
48 



a Andromedae (Alplierat), 
7 Pegasi (Algenib) 



Right Ascen- 
sion. 



2,0 

2.7 



Nebula in Andromeda. 
j3 Ceti 

S3 Andromedae 

a Ursa? Minoris (Polaris). 
a Eridani (Acliernar). . . . 

a Arietis 

Cluster in Perseus. . . . 

o Ceti (Mira) 

a Ceti 

(3 Persei (Algol) 

a Persei 

7] Tauri (Pleiades) 

y Tauri (Hyades) 

a Tauri (Aldebaran) 

a Auriga? (Capella) 

(3 Orionis (Rigel) 

6 Orionis 

Nebula in Orion 

a Orionis 

a Argus (Canopus) 

a Canis Maj. (Sirius). . . . 
a Geminorum (Castor). . . 
a Canis Min. (Procyon). . 
;3 Geminorum (Pollux) . . 

Cluster, Prsesepe 

a Hydra? , 

a Leonis (Pegulus) 

7] Argus (variable) 

a Ursa? Maj oris (Dublie) 

a Crucis 

a Virginis (Spica) 

a Bootis (Arcturus) 

i3 Ursa? Minoris 

a Corona? Borealis 

a Scorpii (Antares) 

a Ophiuclii 

a Lyra? (Vega) 

Annular Nebula, Lyra 

a Aquila? (Altai r) 

Dumb-bell Nebula . . . 

a Delphini 

a Cvgni 

12 tear Catalogue, 1879. 

61 Cygni 

a Piscis Aus.(Fomalhaut) 
a Pegasi (Markab) 



2, 

2.3 

2. 

1. 

2, 



Var. 
2.3 

Var. 

2. 

3*. 

4. 

1. 

1. 

1. 
Var. 



Var. 
1. 
1. 
1.7 
1. 
1.3 

2. 
13 
1-6 
2. 
1. 
1. 
1. 
2. 
2. 
1.3 
2. 
1. 

'l.3 



3.7 

1.7 

6. 

5. 

1.3 

2. 



H. II. S. 

2 26 
7 18 
35 

37 49 

1 3 17 
16 37 
33 25 

41 
10 
13 
56 16 

41 
3 16 7 

3 40 38 

4 13 15 
29 19 

8 11 

9 
26 8 
29 

48 56 
21 24 

6 40 4 

7 27 15 
7 33 16 

7 38 16 

8 20 

9 21 56 
10 2 14 
10 40 36 
10 56 37 

12 20 11 

13 19 8 

14 10 25 

14 51 3 

15 29 49 

16 22 21 

17 29 36 

18 33 3 

18 49 

19 45 10 

19 54 

20 34 18 
20 37 31 

20 52 46 

21 1 45 

22 51 17 
22 59 2 



Annual 
Var. 



+ 3.09 
+ 3.08 

+ 3.01 
+ 3.34 
f 22.46 
+ 2.23 
+ 3.37 



North Polar 
Distance. 



+ 3.12 | 

+ 3.88 
+ 4.25 
+ 3.55 
+ 3.40 
+ 3.43 
+ 4.42 
+ 2.88 
+ 3.06 

+ 3.24 
+ 1.33 

+ 2.64 
+ 3.84 
+ 3.14 
+ 3.67 



+ 2.94 
+ 3.20 
+ 2.31 
+ 3.75 
+ 3.27 
+ 3.15 
+ 2.73 
—0.23 
+ 2.54 
+ 3.67 
+ 2.78 
+ 2.03 



+ 2.92 



+ 2.79 
+ 2.04 
-2.53 

+ 2.68 
+ 3.32 
+ 2.98 



61 32 40 

75 27 21 
49 23 

108 37 5 

64 59 22 

1 18 16 

147 49 16 
67 4 55 
33 

93 32 
86 21 43 
49 29 18 
40 32 57 
66 15 6 
74 39 3 
73 43 22 
44 7 14 
98 20 
90 23 
95 29 
82 36 

142 37 

106 33 33 
57 51 37 
84 28 52 

61 41 50 

69 50 

98 9 38 

77 28 16 

149 4 48 

27 37 42 

152 27 42 

100 33 38 

70 13 7 
15 22 28 

62 53 52 
116 10 32 

77 21 20 
51 19 22 
57 7 
81 26 5 
67 36 

74 29 36 
45 7 49 

9 52 47 

51 48 56 

120 13 53 

75 24 48 



Annual 
Var. 



-19.89 
-20.02 



-19.80 
-19.17 
-18.94 
-18.36 
-17.18 



-14.32 

-14.13 

13.11 

-11.40 

-8.98 

-7.53 

-4.06 

-4.42 

-2.94 

-6.97 

+ 1.88 
+ 4.69 
+ 7.53 
+ 8.97 
+ 8.40 



+ 15.44 

+ 17.46 

+ 18.86 

+ 19.35 

+ 20.01 

+ 18.91 

+ 18.89 

+ 14.72 

+ 12.32 

+ 8.32 

+ 2.89 

-3.15 



9.25 



-12.50 
-12.71 
-13.70 
-17.52 
-18.99 
-19.30 



PLANETOIDS. 



231 



Table V. The Planetoids. 



No. 



10 

11 

12 

13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 



NAME. 



Ceres , 

Pallas 

Juno 

Vesta 

Astrsea 

Hebe 

Iris 

Flora 

Metis 

Hygeia 

Parthenope. . 

Victoria 

Egeria 

Irene 

Eunomia 

Psyche 

Thetis 

Melpomene. . 

Fortuna 

Massalia 

Lutetia 

Calliope 

Thalia 

Themis 

Phocea 

Proserpina . . 
Euterpe 

Bellona 

Amphitrite.. . 

Urania 

Euphrosyne . 

Pomona 

Polyhymnia.. 

Circe 

Leucothea . . . 

Atalanta 

Fides 

Leda 

Laetitia 

Harmonia 

Daphne 

Isis 

Ariadne 

Nysa 

Eugenia . 

Hestia 

Aglaia 

Doris 

Pales 

Virginia 



Mean daily *£** 
motion 



mean 
dist'nce 



770.8332 
769.7324 
812.9059 
976.7787 
857.9269 
939.3696 
962.5806 

1086.3309 
962.3390 
637.1610' 
923.6604i 
994.8347J 
857.9451! 
852.4385; 
825.4550 
710.9629J 
911.3975! 

1020.1198 
9296590; 
949.0444 
933.5544 
715.6529! 
833.0737 
640.1662 
954.6367 
819.6847: 
9866944' 
766.0691 ' 
869.0352 
975.1642 
635.1686 
852.5880; 
732.0291' 
806.1634 
685.1834 
780.0110 
826.0660 
782.5641 
769.9967 

1039.3353 
770.1514 
930.9057 

1084.1384 
941.3988 
789.0034 
883.9660 
725.9827 
646.1069 
653.3922 
822.4986 



.442031 
.442444 
.426644 
.373474 
.411037 
.384780 
.377713 
.342696 
.377786 
.497171 
.389663 
.368139 
.411031 
.412896 
.422209 
.465440 
.393532 
.360903 
.387788 
.381813 
.386578 
.463536 
.419548 
,495809 
380112 
424240 
,370549 
,443825 
.402312 
,373952 
498079 
412845 
,456985 
,429055 
476133 
438604 
,421994 
437657 
.442345 
,355500 
.442287 
.387401 
.343281 
384155 
.435285 
.402381 



.493134 



,423247 



No. 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 
61 
62 
63 
64 
65 
66 
67 



70 

71 

2 

73 

74 
75 
76 
77 
78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
90 
91 
92 
93 
94 
95 
96 
97 
98 
99 
100 



NAME. 



Nemausa , . . 

Europa 

Calypso 

Alexandra. . 

Pandora 

Melete 

Mnemosyne. 
Concordia . . 

Elpis 

Echo 

Danae 

Erato 

Ausonia. . . . 
Angelina. . . 

Cybele 

Maia 

Asia 

Leto 

Hesperia 

Panopea. . . . 

Niobe 

Feronia 

Clytia 

Galatea 

Eurydice . . . 

Freia 

Frigga 

Diana. 

Eurynome. . 

Sappho 

Terpsichore. 
Alcmene. . . . 

Beatrice 

Clio 

Io 

Semele 

Sylvia 

Thisbe 

Julia 

Antiope 

iEgina 

Undina 

Minerva .... 

Aurora 

Arethusa. . . 

Mgle 

Clotho 

Ianthe 

Dice 

Hecate 



Mean daily- 
motion 



975.6485 
651.2204 
837.8551 
794.1220 
774.3196 
847.7131 
635.2707 
799.5964 
793.9788 
958.1112 
687 6656 
642.5658 
956.1364 
806.8077 
558.3014 
824.7087 
941.5410 
765.2766 
689.8760 
839.0994 
774.6491 

1040.1026 
815.4003 
765.7921 
813.0315 
5609129 
814.1350 
836.5607 
928.8736 

1020 0052 
736.1744 
772.7477 
936.6007 
977.8108 
821. 4080 ; 
649.2352 ! 
543.7017! 
770.2917J 
870.8412 
636.1509J 
851.2296 1 
622.3687J 
775.6388 
630.8636^ 
659.2278 ! 
666.2189 
813.1887 
805.3700 
758.6620 
652.0664 



Log. of 

mean 

dist'nce. 



.373809 
.490852 
.417891 
.433412 
.440724 
.414505 
.498032 
.431423 
.433465 
.379060 
.475086 
.494726 
.379658 
.428824 
.535425 
.422471 
.384111 
.444125 
.474157 
.417462 
.440601 
.355287 
.425757 
.443930 
.426599 
.534074 
.426206 
.418339 
.388033 
.360936 
.455350 
.441312 
.385635' 
.373167 
.423632 
.491736 
.543097 
.442234 
.406712 
.497631 
.413306 
.503972 
.440231 
.500047 
.487314 
.484260 
.426543 
.429341 
.446640 
.490476 



232 



PLANETOIDS. 



Table V. (continued). The Planetoids. 



No. 



NAME. 



101 
102 
103 

104 | 

105 I 
106! 
107 ; 
108! 
109 
110 J 
1111 

112 j 

113 j 

114 i 
115 
116 
117 

118 1 

119 i 

120 | 
121 
122! 

123 1 

124 I 

125 I 

126 I 
127 
128 
129! 
130! 

131 i 

132 i 

133 I 

134 i 

135 i 

136 1 

137 1 

138 | 
139, 
140! 
141 
142 
143 ! 
144 
145 
146 
147 
148 
149 
150 



Helena 

Miriam 

Hera 

Clymene 

Artemis 

Dione 

Camilla 

Hecuba 

Felicitas. . . . 

Lydia 

Ate 

lpkigenia.. . . 
Amalthea. . . . 
Cassandra . . . 

Thyra 

Sirona 

Lomia 

Peith<» 

Althea 

Lachesis 

Hermone. . . . 

Gerda 

Brunhild 

Alcestis 

Liberatrix. . . 

Velleda 

Johanna 

Nemesis 

Antigone .... 

Electra 

Vala 

^Ithra 

Cyrene 

Sophvosyne. . 

Hertha 

Austria 

Melibcea 

Tolosa 

Juewa 

Siwa 

Lumen 

Polana 

Adria 

Vibilia 

Adeona 

Lucina 

Protogeneia. . 

Gallia 

Medusa. 

Nuwa 



Mean daily 
motion 



853.6127 
816.7370 
799.0675 
634.4466 
971.0795 
629.5650 
545.4463 
616.3698 
802.0510 
785.1449 
849.9278 
934.4391 
968.1836 
810.8275 
965.9609 
771.4040 
686.0326 
931.6917 
855.5046 
644.3548 
551.5624 
615.5690 
801.8499 
832.0020 
780.7231 
930.9792 
775.3364 
777.4964 
727.2294 
642.9388 
942.2999 
846.3646 
663.5850 
8645740 
638.1149 

1026.3921 
641.8566 
926.0192 
765.7567 
789.1234 
814.5161 
942.8756 
773.0080 
821.2984 1 
815.4470! 
789.8850 
638.6654! 
769.5145 

1139.1950 
689.3407, 



Log. of 
mean No. 
dist'nce. 



412497 
.425283 
.431615 
.498408 
.375168 
.500644 
.542170 
.506777 
.430555 
.436704 
.413749 
.386304 
.376032 
.427385 
.376698 
.441816 
.475775 
.387156 
.411856 
.493921 
.538941 
.507153 
.430609 
.419921 
.438339 
.387377! 
.440344 ; 
.439538! 
458890 
.494558; 
.3838781 
.414966 
.485406 ! 
.408803 
.385167 ; 
.359129 
.495046 
.388924 i 
.443944! 
.436343 
.426071 ' 
.383701! 
.441216 
.423670 
.425740 
.434962 
.496488 
.442526 
.328939 
.474381 



151 
152 
153 
154 
155 
156 
157 
158 
159 
160 
161 
162 
163 
164 
165 
166 
167 
168 
169 
170 
171 
172 
173 
174 
175 
176 
177 
178 
179 
180 
181 
182 
183 
184 
185 
186 
187 
188 
189 
190 
191 
192 
193 
194 
195 
196 
197 
198 
199 
200 



NAME. 



Abundantia. 

Atala 

Hilda 

Bertha 

Scylla 

Xantippe. . . 

Dejanira 

Coronis 

iEmilia 

Una 

Athor 

Laurentia . . 

Erigone 

Eva 

Loreley 
Rhodope . . . 

Urda 

Sibylla 

Zelia 

Myrrha 

Ophelia 

Baucis 



mo 

Phsedra 

Andromache. 

Idunna 

Irma 

Belisaria 

Clytemnestra. 
Garumna. . . . 

Eucharis 

Elsbeth 

Istria 

Deipeia 

Eunike 

Celuta 

Lamberta. 

Menippe 

Phthia , 

Ismene 

Kolga 

Nausika 

Ambrosia 

Proene 

Eurycleia 

Philomela. . . . 

Arete 

Ampella 

Byblis 

Dynamene. . . . 



Mean daily 
motion 



Log. of 

mean 

distance. 



850.7264 

639.0187 

451.5802 

622.3629 

713.7875 1 

670.2300' 

854.8040' 

730.5502 

647.7291! 

787.1915 

970.0005 

673.1350 

981.1480! 

829.6880' 

642.0938J 

803.0021 

614.4750. 

570.0346 

978.5025 

868.8279 ! 

635.5487 

966.3982 

780.2369 

732.1255 

541.0099 

622.6360 

774.6923 

920.0970 

692,2257 

787.4120 

644.0102 

944.0487 

756.3767 

623.2669 

7830772! 

977. 1085 ; 

782.3914: 

748.8250 

924.9882' 

454.0674 

722.4983 

952.5933 

858.2960^ 

836.9383 

728.9100 

653.8370! 

780.9746 

922.9325' 

618.17301 

783.26091 



.413478 
.496329 
.596847 
.503975 
.464292 
.482522 
.412092 
.457571 
.492411 
.435951 
.375489 
.481270 
.372181 
.420728 
.494938 
.430193 
!.507668 
.529402 
.372963 
.407382 
.497905 
.376567 
.438520 
.456947 
.544534 
.503848 
.440585 
.390782 
.473172 
.435870 
.494075 
.383341 
.447526 
.503555 
.437468 
.373376 
.437722 
.450417 
.389247 
.595257 
.460780 
.380673 
.410913 
.418209 
.458222 
.489692 
.438246 
.389894 
.505931 
.437400 



PLANETOIDS. 



233 



Table V. (continued). The Planetoids. 



No. 



NAME. 



201 
202 
203 
204 
205 
206 
207 
208 
209 
210 
211 
212 
213 
214 
215 
216 
217 



Penelope . 
Chryseis. . . 
Pompeia. . . 

Callisto 

Martha . . 

Hersilia 

Hedda. . . . 
Lacrimosa. 

Dido 

Isabella... . 

Isolda 

Medea. . . . 

Lilaea 

Aschera. . . 

(Enone 

Cleopatra. . 
Eudora 



Mean daily 
motion 



809.9320 
655.0080 
782.7813 
812.0185 
766.6919 



1027.3643 
729.1020 
637.0860 
780.0227 
667.2952 
644.9370 
779.8090 
840.9460 
770.4950 
759.6820 
665.7647 



Log. of 

mean 

dist'nce. 

.427706 
.489173 
.437577 
.426960 
.443590 



,358855 
.458146 
497206 
,438599 
,483792 
,493660 
438679 
416826 
,442158 
446250 
,484457 



No. 



218 
219 
220 
221 
222 
223 
224 
225 
226 
227 
228 
229 
330 
231 
232 
233 



NAME. 



Bianca 

Thusnelda. .. 
(Mar. 19, 1881 
(Jan. 18, 1882 



Philosophia. . 



Athamantis. . 
(Sept, 10, 1882 

Russia 

(May 11, 1883 



Mean daily 
motion 



817.2760 
982.3480 
974.5910 
678.2950 
645.2880 
650.1600 
826.1800 
568.9810 
792.4160 
626.1270 
1084.5100 
567.8920 
963.8230 
701.3150 
870.2300 



Log. of 

mean 

dist'nce. 

.425090 
.371828 
.374123 
.479058 
.493502 
.491324 
.421954 
.529939 
.434036 
.502229 
,343182 
.530494 
,37734a 



406915 



PLATE II 

COMET OF 1858,-NKBULi:. 




PLATE 111. 

PART OF GALAXY.-DOUBLE STARS 




I. Castor. 2. y Leonis. 


3. 39 Drac. 


4. A Oph. 5. 11 Monoc. 


6. ; Cancri 


HIB 




S ■ 


B 


H 



Revolutions of y Virginia. 




1837. 1838. 1839. 1840. 1845. 1850. 1860. Orbit 



PLATE IV 

CLUSTERS.-NEBULH. 




